muHVT: Predicting Cells with Layers using predictLayerHVT

Zubin Dowlaty, Srinivasan Sudarsanam, Somya Shambhawi

2023-06-23

1 Abstract

The muHVT package is a collection of R functions to facilitate building topology preserving maps for rich multivariate data analysis. Tending towards a big data preponderance, a large number of rows. A collection of R functions for this typical workflow is organized below:

  1. Data Compression: Vector quantization (VQ), HVQ (hierarchical vector quantization) using means or medians. This step compresses the rows (long data frame) using a compression objective.

  2. Data Projection: Dimension projection of the compressed cells to 1D,2D or 3D with the Sammons Non-linear Algorithm. This step creates topology preserving map (also called as embedding) coordinates into the desired output dimension.

  3. Tessellation: Create cells required for object visualization using the Voronoi Tessellation method, package includes heatmap plots for hierarchical Voronoi tessellations (HVT). This step enables data insights, visualization, and interaction with the topology preserving map. Useful for semi-supervised tasks.

  4. Prediction: Scoring new data sets and recording their assignment using the map objects from the above steps, in a sequence of maps if required.

2 Example: muHVT with the Personal Computer dataset

Data Understanding

In this vignette, we will use the Prices of Personal Computers dataset. This dataset contains 6259 observations and 6 features. The dataset observes the price from 1993 to 1995 of 486 personal computers in the US. The variables are price, speed, hd, ram, screen and ads.

Here, we load the data and store into a variable computers.

set.seed(240)
# Load data from csv files
computers <- read.csv("https://raw.githubusercontent.com/Mu-Sigma/muHVT/master/vignettes/sample_dataset/Computers.csv")

Raw Personal Computers Dataset

The Computers dataset includes the following columns:

Let’s explore the Personal Computers Dataset containing (6259 points). For the shake of brevity we are displaying first six rows.

Table(head(computers), scroll = T, limit = 20)
X price speed hd ram screen cd multi premium ads trend
1 1499 25 80 4 14 no no yes 94 1
2 1795 33 85 2 14 no no yes 94 1
3 1595 25 170 4 15 no no yes 94 1
4 1849 25 170 8 14 no no no 94 1
5 3295 33 340 16 14 no no yes 94 1
6 3695 66 340 16 14 no no yes 94 1

Now, let us check the structure of the data and analyse its summary.

str(computers)
#> 'data.frame':    6259 obs. of  11 variables:
#>  $ X      : int  1 2 3 4 5 6 7 8 9 10 ...
#>  $ price  : int  1499 1795 1595 1849 3295 3695 1720 1995 2225 2575 ...
#>  $ speed  : int  25 33 25 25 33 66 25 50 50 50 ...
#>  $ hd     : int  80 85 170 170 340 340 170 85 210 210 ...
#>  $ ram    : int  4 2 4 8 16 16 4 2 8 4 ...
#>  $ screen : int  14 14 15 14 14 14 14 14 14 15 ...
#>  $ cd     : chr  "no" "no" "no" "no" ...
#>  $ multi  : chr  "no" "no" "no" "no" ...
#>  $ premium: chr  "yes" "yes" "yes" "no" ...
#>  $ ads    : int  94 94 94 94 94 94 94 94 94 94 ...
#>  $ trend  : int  1 1 1 1 1 1 1 1 1 1 ...
summary(computers)
#>        X            price          speed              hd        
#>  Min.   :   1   Min.   : 949   Min.   : 25.00   Min.   :  80.0  
#>  1st Qu.:1566   1st Qu.:1794   1st Qu.: 33.00   1st Qu.: 214.0  
#>  Median :3130   Median :2144   Median : 50.00   Median : 340.0  
#>  Mean   :3130   Mean   :2220   Mean   : 52.01   Mean   : 416.6  
#>  3rd Qu.:4694   3rd Qu.:2595   3rd Qu.: 66.00   3rd Qu.: 528.0  
#>  Max.   :6259   Max.   :5399   Max.   :100.00   Max.   :2100.0  
#>       ram             screen           cd               multi          
#>  Min.   : 2.000   Min.   :14.00   Length:6259        Length:6259       
#>  1st Qu.: 4.000   1st Qu.:14.00   Class :character   Class :character  
#>  Median : 8.000   Median :14.00   Mode  :character   Mode  :character  
#>  Mean   : 8.287   Mean   :14.61                                        
#>  3rd Qu.: 8.000   3rd Qu.:15.00                                        
#>  Max.   :32.000   Max.   :17.00                                        
#>    premium               ads            trend      
#>  Length:6259        Min.   : 39.0   Min.   : 1.00  
#>  Class :character   1st Qu.:162.5   1st Qu.:10.00  
#>  Mode  :character   Median :246.0   Median :16.00  
#>                     Mean   :221.3   Mean   :15.93  
#>                     3rd Qu.:275.0   3rd Qu.:21.50  
#>                     Max.   :339.0   Max.   :35.00

Let us first split the data into train and test. We will randomly select 80% of the data for training and remaining as testing.


num_rows <- nrow(computers)
set.seed(123)
train_indices <- sample(1:num_rows, 0.8 * num_rows)
trainComputers <- computers[train_indices, ]
testComputers <- computers[-train_indices, ]

K-means is not suitable for factor variables as the sample space for factor variables is discrete. A Euclidean distance function on such a space isn’t really meaningful. Hence, we will delete the factor variables(X, cd, multi, premium, trend) in our dataset.

trainComputers <-
  trainComputers %>% dplyr::select(-c(X, cd, multi, premium, trend))
testComputers <-
  testComputers %>% dplyr::select(-c(X, cd, multi, premium, trend))

Raw Training Dataset

Now, lets have a look at the randomly selected raw training dataset containing (5007 data points). For the sake of brevity we are displaying first six rows.

trainComputers_data <- trainComputers %>% as.data.frame() %>% round(4)
trainComputers_data$Row.No <- as.numeric(row.names(trainComputers_data))
trainComputers_data <- trainComputers_data %>% dplyr::select(Row.No,price,speed,hd,ram,screen,ads)
row.names(trainComputers_data) <- NULL
Table(head(trainComputers_data))
Row.No price speed hd ram screen ads
2463 2799 50 230 8 15 216
2511 2197 33 270 4 14 216
2227 2744 50 340 8 17 275
526 2999 66 245 16 15 139
4291 1974 33 200 4 14 248
2986 2490 33 528 16 14 267

Raw Testing Dataset

Now, lets have a look at the randomly selected raw testing dataset containing (1252 data points). For the sake of brevity we are displaying first six rows.

#testComputers <- scale(testComputers, center = scale_attr$`scaled:center`, scale = scale_attr$`scaled:scale`) 
testComputers_data <- testComputers %>% as.data.frame() %>% round(4)
testComputers_data$Row.No <- as.numeric(row.names(testComputers_data))
testComputers_data <- testComputers_data %>% dplyr::select(Row.No,price,speed,hd,ram,screen,ads)
rownames(testComputers_data) <- NULL
Table(head(testComputers_data))
Row.No price speed hd ram screen ads
3 1595 25 170 4 15 94
4 1849 25 170 8 14 94
7 1720 25 170 4 14 94
10 2575 50 210 4 15 94
11 2195 33 170 8 15 94
14 2295 25 245 8 14 94

3 Map A : Base Compressed Map

Let us try to visualize the compressed Map A from the flow diagram below.

Figure 1: Flow map with highlighted bounding box in red around compressed map A

Figure 1: Flow map with highlighted bounding box in red around compressed map A

This package can perform vector quantization using the following algorithms -

For more information on vector quantization, refer the following link.

The HVT function constructs highly compressed hierarchical Voronoi tessellations. The raw data is first scaled and this scaled data is supplied as input to the vector quantization algorithm. The vector quantization algorithm compresses the dataset until a user-defined compression percentage/rate is achieved using a parameter called quantization error which acts as a threshold and determines the compression percentage. It means that for a given user-defined compression percentage we get the ‘n’ number of cells, then all of these cells formed will have a quantization error less than the threshold quantization error.

Let’s try to comprehend the HVT function first before moving ahead.

HVT(
  dataset,
  min_compression_perc,
  n_cells,
  depth,
  quant.err,
  distance_metric = c("L1_Norm", "L2_Norm"),
  error_metric = c("mean", "max"),
  quant_method = c("kmeans", "kmedoids"),
  normalize = TRUE,
  diagnose = FALSE,
  hvt_validation = FALSE,
  train_validation_split_ratio = 0.8
)

Each of the parameters have been explained below :

We will use the HVT function to compress our data while preserving essential features of the dataset. Our goal is to achieve data compression upto atleast 80%. In situations where the compression ratio does not meet the desired target, we can explore adjusting the model parameters as a potential solution. This involves making modifications to parameters such as the quantization error threshold or increasing the number of cells and then rerunning the HVT function again.

In our example we will iteratively increase the number of cells until the desired compression percentage is reached instead of increasing the quantization threshold because it may reduce the level of detail captured in the data representation

First, we will construct map A by using the below mentioned model parameters.

3.0.1 Iteration 1:

We will pass the below mentioned model parameters along with training dataset to HVT function.

Model Parameters

set.seed(240)
map_A <- list()
map_A  <- muHVT::HVT(
  trainComputers,
  n_cells = 200,
  depth = 1,
  quant.err = 0.2,
  projection.scale = 10,
  normalize = T,
  distance_metric = "L1_Norm",
  error_metric = "max",
  quant_method = "kmeans"
)

Let’s checkout the compression summary.

compressionSummaryTable(map_A[[3]]$compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 200 83 0.42 n_cells: 200 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 42% of cells have reached the quantization threshold error. Therefore we can further subdivide the cells by increasing the n_cells parameters and then see if desired compression (80%) is reached

3.0.2 Iteration 2:

Since, we are yet to achive atleast 80% compression. Let’s try to compress again using the below mentioned set of model parameters.

Model Parameters

map_A <- list()
map_A <-muHVT::HVT(trainComputers,
                n_cells = 440,
                quant.err = 0.2,
                depth = 1,
                distance_metric = "L1_Norm",
                error_metric = "max",
                quant_method = "kmeans",
                normalize = T)

As per the manual, map_A[[3]] gives us detailed information about the hierarchical vector quantized data. map_A[[3]][['summary']] gives a nice tabular data containing no of points, Quantization Error and the codebook.

The datatable displayed below is the summary from map A

summaryTable(map_A[[3]]$summary,scroll = T,limit = 500)
Segment.Level Segment.Parent Segment.Child n Cell.ID Quant.Error price speed hd ram screen ads
1 1 1 7 46 0.08 -0.76 -0.89 -0.88 -0.76 -0.67 1.57
1 1 2 10 108 0.08 -0.80 -0.89 -0.16 -0.76 -0.67 0.67
1 1 3 15 223 0.12 0.37 -0.89 -0.72 -0.05 0.43 -1.65
1 1 4 11 54 0.07 -1.50 -0.89 -0.75 -0.76 -0.67 0.62
1 1 5 8 146 0.13 -0.31 0.68 -0.95 -0.89 -0.67 -0.14
1 1 6 11 150 0.16 -0.66 0.68 -0.78 -0.79 -0.67 -0.73
1 1 7 11 170 0.1 0.03 -1.24 -0.13 -0.05 -0.67 0.38
1 1 8 8 334 0.15 0.62 2.30 0.08 -0.05 0.43 0.04
1 1 9 8 114 0.07 -0.16 0.68 -1.19 -1.11 -0.67 0.87
1 1 10 7 248 0.17 0.51 -0.08 0.34 -0.05 -0.67 -0.30
1 1 11 9 140 0.12 -0.01 0.68 -1.15 -1.00 -0.67 0.33
1 1 12 7 219 0.14 -1.36 0.24 0.46 -0.05 -0.67 -0.74
1 1 13 9 271 0.05 -1.08 0.68 0.49 -0.05 0.43 -0.84
1 1 14 19 109 0.06 -0.31 -0.89 -0.74 -0.76 -0.67 0.38
1 1 15 6 176 0.08 -0.72 -0.89 -0.07 -0.05 0.43 0.72
1 1 16 17 332 0.14 0.42 2.30 0.10 -0.05 0.43 1.50
1 1 17 12 18 0.05 -1.21 -1.27 -1.19 -1.11 -0.67 0.97
1 1 18 19 149 0.16 -0.68 -0.08 -0.46 -0.76 0.43 0.79
1 1 19 17 428 0.35 0.18 2.30 2.53 1.37 0.43 -2.22
1 1 20 20 320 0.36 0.82 -0.16 -0.09 -0.12 2.64 0.71
1 1 21 3 305 0.18 2.27 -0.35 -0.01 -0.05 -0.67 -1.32
1 1 22 7 227 0.1 -0.51 -0.89 0.45 -0.05 0.43 -0.47
1 1 23 10 178 0.12 0.00 0.68 -0.86 -0.76 -0.67 -0.90
1 1 24 9 365 0.1 0.68 -0.08 1.20 1.37 0.43 -0.36
1 1 25 5 14 0.11 -1.99 -0.89 -0.96 -1.11 -0.67 0.18
1 1 26 3 411 0.05 1.25 -0.89 2.29 2.79 0.43 0.57
1 1 27 18 122 0.15 -0.18 -0.98 -0.85 -0.76 0.43 0.68
1 1 28 15 189 0.11 0.40 -0.92 0.03 -0.05 -0.67 0.87
1 1 29 11 107 0.11 -0.49 -0.96 -0.88 -0.76 -0.67 -0.64
1 1 30 7 423 0.47 3.55 0.12 2.51 1.37 -0.67 0.44
1 1 31 14 90 0.05 -0.63 -0.89 -0.79 -0.76 -0.67 0.58
1 1 32 22 430 0.24 0.63 0.75 3.07 2.79 0.43 -2.27
1 1 33 5 390 0.3 1.37 -0.89 3.73 -0.19 -0.45 0.70
1 1 34 25 101 0.18 -0.85 -0.97 -0.71 -0.76 0.43 0.84
1 1 35 11 425 0.07 0.15 2.30 1.70 1.37 0.43 -2.39
1 1 36 10 358 0.05 0.24 -0.89 1.20 1.37 0.43 -0.84
1 1 37 16 166 0.11 0.03 0.68 -1.08 -0.78 -0.67 -1.65
1 1 38 13 45 0.05 -0.91 -0.89 -1.19 -1.11 -0.67 0.42
1 1 39 8 383 0.12 1.15 2.30 0.45 1.37 -0.67 -0.16
1 1 40 5 9 0.07 -1.24 -0.97 -1.19 -1.11 -0.67 1.57
1 1 41 11 419 0.06 1.41 -0.89 2.29 2.79 0.43 -0.82
1 1 42 8 242 0.14 -0.81 -0.08 0.30 -0.05 0.43 -0.65
1 1 43 13 179 0.09 0.41 0.68 -0.76 -0.76 -0.67 0.40
1 1 44 5 375 0.04 0.06 -0.08 1.70 1.37 0.43 -0.79
1 1 45 20 129 0.14 -1.15 0.68 -0.79 -0.76 -0.67 -0.40
1 1 46 10 292 0.22 0.86 0.68 -0.65 -0.12 0.43 -1.41
1 1 47 5 79 0.12 -0.89 -1.04 -0.94 -0.05 -0.67 1.02
1 1 48 23 246 0.11 -0.42 0.68 0.46 -0.05 -0.67 -0.63
1 1 49 8 207 0.25 0.74 -0.89 -0.40 -0.40 0.43 0.52
1 1 50 11 27 0.06 -1.06 -1.27 -1.19 -1.11 -0.67 0.43
1 1 51 8 51 0.11 -1.23 -0.08 -1.05 -0.85 -0.67 0.94
1 1 52 19 288 0.09 0.81 -0.89 0.45 1.37 -0.67 0.88
1 1 53 7 154 0.1 -0.62 0.68 -0.15 -0.76 -0.67 0.84
1 1 54 10 261 0.15 0.61 -0.08 -0.67 -0.05 0.43 -1.40
1 1 55 10 195 0.15 0.18 0.77 -0.09 -0.76 -0.67 0.83
1 1 56 14 250 0.09 0.52 0.68 -0.69 -0.05 -0.67 -1.61
1 1 57 20 331 0.2 -0.63 0.68 0.30 -0.76 2.64 -0.95
1 1 58 9 379 0.15 1.33 -0.08 -0.65 -0.29 2.64 -1.52
1 1 59 14 11 0.21 -0.28 -1.05 -0.79 -0.76 2.64 0.53
1 1 60 29 359 0.13 1.36 0.68 0.18 1.37 0.43 0.77
1 1 61 6 337 0.1 2.46 0.68 0.21 -0.05 -0.67 -0.87
1 1 62 6 1 0.17 -0.17 -1.21 -1.02 -0.76 2.64 1.32
1 1 63 28 243 0.33 -0.33 2.30 -0.23 -0.46 -0.67 -0.88
1 1 64 8 274 0.07 0.41 -1.27 0.45 1.37 -0.67 0.68
1 1 65 13 362 0.14 1.07 0.75 0.35 1.37 0.43 1.34
1 1 66 10 143 0.07 -0.34 -0.89 -0.80 -0.05 -0.67 -1.66
1 1 67 4 265 0.05 -0.55 0.68 1.23 -0.05 -0.67 -0.69
1 1 68 11 13 0.15 -0.83 -0.89 -0.25 -0.76 2.64 -0.33
1 1 69 8 298 0.17 -0.62 0.20 2.29 -0.05 -0.67 -0.95
1 1 70 4 335 0.06 1.34 -0.08 0.45 1.37 -0.67 -0.08
1 1 71 20 204 0.16 0.09 -0.08 0.02 -0.05 -0.67 0.86
1 1 72 10 42 0.06 -1.49 -0.89 -0.75 -0.76 -0.67 1.04
1 1 73 1 429 0 3.08 0.68 0.04 4.20 0.43 0.71
1 1 74 14 186 0.14 -0.79 -0.89 0.45 -0.05 -0.67 -0.68
1 1 75 4 410 0.37 2.27 0.68 3.73 -0.23 -0.40 0.68
1 1 76 9 163 0.16 1.05 -0.89 -0.41 -0.60 -0.67 0.61
1 1 77 10 400 0.07 -0.03 0.68 1.70 1.37 0.43 -2.38
1 1 78 6 275 0.18 1.14 0.68 0.13 -0.05 -0.67 -0.18
1 1 79 25 241 0.16 -0.88 0.68 0.40 -0.05 -0.67 -1.06
1 1 80 6 245 0.14 -1.22 0.68 -0.30 -0.05 0.43 -0.91
1 1 81 21 120 0.16 -0.46 -0.08 -0.82 -0.62 -0.67 0.70
1 1 82 11 40 0.18 -0.93 -0.99 -1.19 -1.11 0.43 0.37
1 1 83 9 342 0.28 1.16 0.43 -0.52 -0.68 2.64 1.12
1 1 84 8 286 0.05 -0.72 1.11 0.50 -0.05 0.43 -0.83
1 1 85 8 33 0.1 -1.06 -0.89 -1.18 -1.02 -0.67 -0.99
1 1 86 5 282 0.23 -1.50 0.53 0.19 -0.48 0.43 -2.16
1 1 87 17 137 0.07 -0.31 0.68 -0.78 -0.76 -0.67 0.96
1 1 88 19 168 0.07 -0.08 -0.89 0.05 -0.05 -0.67 1.04
1 1 89 7 291 0.05 1.10 -0.89 0.15 1.37 -0.67 0.36
1 1 90 6 24 0.19 -0.97 -1.02 -1.07 -0.94 0.43 -1.40
1 1 91 4 434 0.59 4.07 1.49 1.29 0.30 2.64 0.07
1 1 92 19 409 0.65 0.92 0.42 1.41 1.37 2.64 -0.60
1 1 93 9 393 0.46 2.04 2.30 0.91 -0.05 -0.31 -0.34
1 1 94 8 22 0.08 -1.66 -1.27 -0.84 -0.76 -0.67 0.84
1 1 95 23 158 0.22 -1.41 0.68 -0.20 -0.70 -0.67 -1.05
1 1 96 11 330 0.22 1.55 0.68 -0.45 -0.31 0.43 -1.59
1 1 97 6 145 0.17 0.49 -0.89 -0.64 -0.76 -0.67 -0.17
1 1 98 12 121 0.15 0.21 -0.89 -0.80 -0.76 -0.67 -1.70
1 1 99 14 329 0.24 2.06 0.63 0.31 -0.25 0.43 0.99
1 1 100 16 299 0.05 1.21 -0.89 0.46 1.37 -0.67 0.86
1 1 101 5 328 0.1 -0.87 1.11 1.33 -0.05 0.43 -1.13
1 1 102 10 278 0.1 0.33 -0.93 0.45 1.37 -0.67 0.02
1 1 103 5 102 0.1 -1.05 0.68 -0.76 -0.76 -0.67 1.27
1 1 104 12 25 0.08 -1.25 -1.05 -0.78 -0.76 -0.67 1.57
1 1 105 9 385 0.25 2.05 0.34 -0.21 -0.05 2.64 1.12
1 1 106 10 231 0.18 -0.02 -0.08 0.02 -0.05 0.43 1.29
1 1 107 5 193 0.14 -0.66 -0.08 -0.33 -0.05 -0.67 -1.01
1 1 108 4 418 0.06 -0.03 0.68 3.07 1.37 0.43 -2.25
1 1 109 4 306 0.16 1.83 0.68 -0.59 -0.05 -0.67 -1.57
1 1 110 5 378 0.15 1.09 0.68 1.20 1.37 0.43 -0.11
1 1 111 2 308 0.07 -0.55 0.68 0.26 1.37 -0.67 -1.25
1 1 112 18 239 0.15 0.44 0.75 0.08 -0.05 -0.67 1.05
1 1 113 11 62 0.06 -1.14 -0.89 -0.81 -0.76 -0.67 0.73
1 1 114 5 169 0.07 -0.26 -0.89 0.47 -0.05 -0.67 1.40
1 1 115 18 415 0.11 0.74 -0.08 2.29 2.79 0.43 -0.93
1 1 116 5 97 0.13 0.23 -1.19 -1.05 -0.76 -0.67 0.77
1 1 117 24 19 0.22 -0.06 2.30 -0.89 -0.89 -0.67 1.18
1 1 118 9 184 0.17 -0.98 0.34 -0.29 -0.05 -0.67 -0.08
1 1 119 11 43 0.12 -0.82 -0.89 -1.04 -0.79 -0.67 -1.65
1 1 120 9 257 0.29 2.17 -0.62 0.10 -0.05 -0.55 0.65
1 1 121 27 155 0.18 0.15 0.68 -0.84 -0.76 -0.67 0.81
1 1 122 20 215 0.11 0.33 -0.08 -0.69 -0.05 -0.67 -1.64
1 1 123 14 348 0.24 0.61 0.74 0.37 1.37 0.43 0.25
1 1 124 13 162 0.12 0.01 -1.01 -0.67 -0.05 -0.67 -1.57
1 1 125 7 366 0.22 1.62 0.35 -0.29 1.37 -0.67 -1.65
1 1 126 9 77 0.02 -0.88 -0.89 -0.78 -0.76 -0.67 0.66
1 1 127 25 253 0.14 0.70 0.68 0.16 -0.05 -0.67 0.61
1 1 128 9 84 0.09 -0.27 -1.27 -0.81 -0.76 -0.67 0.81
1 1 129 3 398 0.04 -0.03 -0.08 2.29 1.37 0.43 -1.98
1 1 130 9 412 0.05 0.90 -0.89 2.29 2.79 0.43 -0.48
1 1 131 8 309 0.06 1.43 -0.89 0.41 1.37 -0.67 0.46
1 1 132 8 267 0.08 0.10 0.68 0.37 -0.05 0.43 1.57
1 1 133 16 252 0.22 0.94 -0.08 -0.40 -0.14 0.43 0.43
1 1 134 15 119 0.13 -0.60 -0.99 -0.75 -0.76 0.43 0.24
1 1 135 13 48 0.09 -0.66 -0.92 -1.19 -1.11 -0.67 0.82
1 1 136 9 326 0.11 1.45 -0.08 0.32 1.37 -0.67 0.38
1 1 137 11 386 0.22 1.37 0.68 -0.66 -0.18 2.64 -1.36
1 1 138 20 255 0.15 0.43 -0.89 -0.34 -0.05 2.64 0.68
1 1 139 8 161 0.14 -0.68 -0.94 -0.08 -0.05 0.43 1.41
1 1 140 10 38 0.13 -1.22 -1.19 -0.92 -0.76 0.43 0.91
1 1 141 14 180 0.12 0.09 0.68 -0.56 -0.76 -0.67 0.07
1 1 142 13 427 0.21 2.01 0.62 2.29 2.79 0.43 -0.18
1 1 143 15 354 0.13 1.63 0.68 0.39 1.37 -0.67 0.26
1 1 144 11 301 0.14 0.82 -1.03 0.15 1.37 -0.67 -0.85
1 1 145 7 205 0.13 0.24 -0.08 -0.05 -0.05 -0.67 1.57
1 1 146 10 327 0.14 0.66 0.68 0.60 1.37 -0.67 0.42
1 1 147 11 403 0.11 0.92 2.30 1.22 1.37 0.43 -0.79
1 1 148 7 127 0.08 -0.20 -0.89 -0.10 -0.76 -0.67 0.69
1 1 149 6 217 0.11 0.41 -0.89 0.26 -0.05 -0.67 -0.34
1 1 150 11 323 0.1 0.96 -0.89 0.46 1.37 0.43 0.72
1 1 151 9 208 0.16 0.16 0.68 -0.60 -0.05 -0.67 0.43
1 1 152 14 405 0.14 -0.04 0.68 2.29 1.37 0.43 -2.24
1 1 153 10 98 0.08 -0.22 -0.89 -1.14 -0.76 -0.67 0.33
1 1 154 7 47 0.05 -0.99 -0.89 -1.16 -0.76 -0.67 1.03
1 1 155 8 182 0.15 -1.00 0.68 0.08 -0.76 -0.67 -0.46
1 1 156 9 153 0.07 -0.24 -0.93 0.04 -0.05 -0.67 1.57
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1 1 415 8 437 0.27 1.02 1.00 3.07 2.79 2.64 -2.30
1 1 416 7 284 0.19 0.20 -0.66 -0.74 -0.05 2.64 -0.71
1 1 417 9 264 0.06 0.29 -0.89 0.45 1.37 -0.67 0.66
1 1 418 21 238 0.21 0.24 -0.08 -0.09 -0.05 0.43 0.69
1 1 419 3 296 0.02 -0.47 0.68 0.03 -0.05 0.43 -2.34
1 1 420 10 100 0.16 -0.44 -0.97 -0.87 -0.76 0.43 1.21
1 1 421 11 174 0.15 -0.61 0.68 -0.02 -0.76 -0.67 0.17
1 1 422 9 311 0.25 1.52 -0.53 -0.55 -0.76 2.64 0.36
1 1 423 6 135 0.09 -0.58 -1.27 0.12 -0.05 -0.67 1.29
1 1 424 5 368 0.15 2.37 0.68 0.36 1.37 -0.67 0.68
1 1 425 9 373 0.12 1.65 0.68 0.39 1.37 0.43 -0.01
1 1 426 10 228 0.13 0.42 -0.08 -0.33 -0.05 -0.67 -0.92
1 1 427 20 86 0.22 -1.23 -0.93 -0.85 -0.83 0.43 -0.46
1 1 428 7 106 0.06 -0.07 -0.89 -0.76 -0.76 -0.67 0.98
1 1 429 8 138 0.12 -0.45 -0.99 -0.78 -0.76 0.43 -0.77
1 1 430 8 30 0.12 -0.51 1.05 -1.19 -1.11 -0.67 1.36
1 1 431 19 357 0.16 1.03 0.68 -0.27 -0.05 2.64 0.61
1 1 432 20 406 0.38 0.02 2.30 0.99 -0.05 2.64 -1.12
1 1 433 10 302 0.26 1.83 0.53 -0.09 -0.33 0.43 0.24
1 1 434 5 73 0.04 -0.72 -0.89 -0.95 -0.76 -0.67 0.82
1 1 435 1 440 0 5.51 0.68 3.07 4.20 2.64 0.50
1 1 436 14 300 0.29 2.04 0.68 0.38 -0.15 -0.67 0.69
1 1 437 19 74 0.1 -1.10 -0.89 -0.81 -0.76 -0.67 -0.90
1 1 438 7 347 0.13 1.21 -0.08 0.41 1.37 0.43 0.59
1 1 439 31 374 0.28 -0.27 0.87 0.49 -0.05 2.64 -1.24
1 1 440 17 233 0.2 -0.48 0.68 -0.10 -0.13 0.43 0.60

Now let us understand what each column in the above table means:

All the columns after this will contain centroids for each cell. They can also be called a codebook, which represents a collection of all centroids or codewords.

Now, let’s check the compression summary for HVT (map A). The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

mapA_compression_summary <- map_A[[3]]$compression_summary %>%  dplyr::mutate_if(is.numeric, funs(round(.,4)))
compressionSummaryTable(mapA_compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 440 355 0.81 n_cells: 440 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 81% of the cells have hit the quantization threshold error.Since we are successfully able to attain the desired compression percentage, so we will not further subdivide the cells

Now let’s try to understand plotHVT function. The parameters have been explained in detail below:

plotHVT(hvt.results, line.width, color.vec, pch1 = 21, centroid.size = 3, title = NULL, maxDepth = 1)

Let’s plot the Voronoi tessellation for layer 1 (map A).

muHVT::plotHVT(map_A,
        line.width = c(0.4), 
        color.vec = c("#141B41"),
        centroid.size = 0.01,
        maxDepth = 1) 
Figure 2: The Voronoi Tessellation for layer 1 (map A) shown for the 440 cells in the dataset ’computers’

Figure 2: The Voronoi Tessellation for layer 1 (map A) shown for the 440 cells in the dataset ’computers’

Heat Maps

We will now overlay all the features as heatmap over the Voronoi Tessellation plot for better visualization and identification of patterns, trends, and variations in the data.

Let’s have a look at the function hvtHmap that we will use to overlay features as heatmap.

hvtHmap(hvt.results, dataset, child.level, hmap.cols, color.vec ,line.width, palette.color = 6)

Now let’s plot the Voronoi Tessellation with the heatmap overlaid for all the features in the torus data for better visualization and interpretation of data patterns and distributions.

The heatmaps displayed below provides a visual representation of the spatial characteristics of the computers data, allowing us to observe patterns and trends in the distribution of each of the features (n,price,speed,hd,ram,screen,ads). The sheer green shades highlight regions with higher values in each of the heatmaps, while the indigo shades indicate areas with the lowest values in each of the heatmaps. By analyzing these heatmaps, we can gain insights into the variations and relationships between each of these features within the computers data.


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "n",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
) 
Figure 3: The Voronoi Tessellation with the heat map overlaid for No. of entities in each cell

Figure 3: The Voronoi Tessellation with the heat map overlaid for No. of entities in each cell


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "price",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
) 
Figure 4: The Voronoi Tessellation with the heat map overlaid for variable ’price’ in the ’computers’ dataset

Figure 4: The Voronoi Tessellation with the heat map overlaid for variable ’price’ in the ’computers’ dataset


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "speed",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
) 
Figure 5: The Voronoi Tessellation with the heat map overlaid for variable ’speed’ in the ’computers’ dataset

Figure 5: The Voronoi Tessellation with the heat map overlaid for variable ’speed’ in the ’computers’ dataset


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "hd",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
) 
Figure 6: The Voronoi Tessellation with the heat map overlaid for variable ’hd’ in the ’computers’ dataset

Figure 6: The Voronoi Tessellation with the heat map overlaid for variable ’hd’ in the ’computers’ dataset


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "ram",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
) 
Figure 7: The Voronoi Tessellation with the heat map overlaid for variable ’ram’ in the ’computers’ dataset

Figure 7: The Voronoi Tessellation with the heat map overlaid for variable ’ram’ in the ’computers’ dataset


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "screen",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
) 
Figure 8: The Voronoi Tessellation with the heat map overlaid for variable ’screen’ in the ’computers’ dataset

Figure 8: The Voronoi Tessellation with the heat map overlaid for variable ’screen’ in the ’computers’ dataset


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "ads",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
) 
Figure 9: The Voronoi Tessellation with the heat map overlaid for variable ’ads’ in the ’computers’ dataset

Figure 9: The Voronoi Tessellation with the heat map overlaid for variable ’ads’ in the ’computers’ dataset

4 Map B : Compressed Novelty Map

Let us try to visualize the Map B from the flow diagram below.

Figure 10: Flow map with highlighted bounding box in red around map B

Figure 10: Flow map with highlighted bounding box in red around map B

In this section, we will manually figure out the novelty cells from the plotted map A and store it in identified_Novelty_cells variable.

Note: For manual selecting the novelty cells from map A, one can enhance its interactivity by adding plotly elements to the code. This will transform map A into an interactive plot, allowing users to actively engage with the data. By hovering over the centroids of the cells, a tag containing segment child information will be displayed. Users can explore the map by hovering over different cells and selectively choose the novelty cells they wish to consider. Added an image for reference.

Figure 11: Manually selecting novelty cells

Figure 11: Manually selecting novelty cells

The removeNovelty function removes the identified novelty cell(s) from the dataset and stores those records separately.

It takes input as the cell number (Segment.Child) of the manually identified novelty cell(s) from the above table and the compressed HVT map (map A). It returns a list of two items: dataset with novelty records, and a subset of the dataset without the novelty records.

identified_Novelty_cells <<- c(73,321,332,338,435)
output_list <- removeNovelty(identified_Novelty_cells, map_A)

[1] “The following cell(s) have been removed as outliers from the dataset: 73 321 332 338 435”

data_with_novelty <- output_list[[1]]
dataset_without_novelty <- output_list[[2]]

Let’s have a look at the data with novelties.For the sake of brevity, we will only show the first 10 rows.

novelty_data <- data_with_novelty
novelty_data$Row.No <- row.names(novelty_data)
novelty_data <- novelty_data %>% dplyr::select("Row.No","Cell.ID","Cell.Number","price","speed","hd","ram","screen","ads")
colnames(novelty_data) <- c("Row.No","Cell.ID","Segment.Child","price","speed","hd","ram","screen","ads")
novelty_data %>% head(100) %>% 
  as.data.frame() %>%
  Table(scroll = T, limit = 20)
Row.No Cell.ID Segment.Child price speed hd ram screen ads
1 429 73 3.0762240 0.6794579 0.0421969 4.2031619 0.4307274 0.7120258
2 438 321 2.6449847 0.6794579 6.5710368 1.3676416 0.4307274 0.6851382
3 438 321 1.6578103 2.2969425 6.5710368 1.3676416 0.4307274 -1.2507722
4 439 332 0.9130997 0.6794579 4.6232922 2.7854017 0.4307274 -2.4607161
5 439 332 1.8569770 1.1076156 4.6232922 4.2031619 2.6404120 -2.4607161
6 439 332 1.5192595 2.2969425 4.6232922 4.2031619 0.4307274 -2.4607161
7 439 332 1.4482522 1.1076156 4.6232922 4.2031619 0.4307274 -2.4607161
8 439 332 1.8569770 1.1076156 4.6232922 4.2031619 2.6404120 -2.4607161
9 439 332 1.1555636 2.2969425 4.6232922 2.7854017 0.4307274 -2.4607161
10 439 332 1.7773103 1.1076156 4.6232922 4.2031619 0.4307274 -2.4607161
11 439 332 1.3460710 0.6794579 4.6232922 4.2031619 0.4307274 -2.4607161
12 439 332 1.4482522 1.1076156 4.6232922 4.2031619 0.4307274 -2.4607161
13 218 338 -1.4959522 0.6794579 0.4473277 -0.7589986 -0.6741149 -2.2993903
14 218 338 -1.5115391 0.6794579 0.4473277 -0.7589986 -0.6741149 -1.9767386
15 218 338 -1.1668940 0.6794579 0.4473277 -0.7589986 -0.6741149 -2.2859465
16 218 338 -1.4959522 0.6794579 0.4473277 -0.7589986 -0.6741149 -2.4472723
17 218 338 -1.5981334 0.6794579 0.4473277 -0.7589986 -0.6741149 -2.2859465
18 218 338 -1.1668940 0.6794579 0.4473277 -0.7589986 -0.6741149 -2.2993903
19 218 338 -1.1668940 0.6794579 0.4473277 -0.7589986 -0.6741149 -2.4472723
20 218 338 -1.3227637 0.6794579 0.4473277 -0.7589986 -0.6741149 -1.9767386

4.1 Voronoi Tessellation with highlighted novelty cell

The plotCells function is used to plot the Voronoi tessellation using the compressed HVT map (map A) and highlights the identified novelty cell(s) in red on the map.

plotCells(identified_Novelty_cells, map_A,line.width = c(0.4),centroid.size = 0.01)
Figure 12: The Voronoi Tessellation constructed using the compressed HVT map (map A) with the novelty cell(s) highlighted in red

Figure 12: The Voronoi Tessellation constructed using the compressed HVT map (map A) with the novelty cell(s) highlighted in red

We pass the dataframe with novelty records to HVT function along with other model parameters mentioned below to generate map B (layer2)

Model Parameters

colnames(data_with_novelty) <- c("Cell.ID","Segment.Child","price","speed","hd","ram","screen","ads")
dataset_with_novelty <- data_with_novelty[,-1:-2]
map_B <- list()
mapA_scale_summary = map_A[[3]]$scale_summary
map_B <- muHVT::HVT(dataset_with_novelty,
                  n_cells = 5,
                  depth = 1,
                  quant.err = 0.2,
                  projection.scale = 10,
                  normalize = F,
                  distance_metric = "L1_Norm",
                  error_metric = "max",
                  quant_method = "kmeans",
                  diagnose = F
                  )

The datatable displayed below is the summary from map B (layer 2).

summaryTable(map_B[[3]]$summary,scroll = T,limit = 500)
Segment.Level Segment.Parent Segment.Child n Cell.ID Quant.Error price speed hd ram screen ads
1 1 1 2 5 0.01 -1.55 -0.08 0.45 -0.76 -0.67 -1.98
1 1 2 9 3 0.49 1.48 1.28 4.62 3.89 0.92 -2.46
1 1 3 2 2 0.38 2.15 1.49 6.57 1.37 0.43 -0.28
1 1 4 9 4 0.08 -1.33 0.68 0.45 -0.76 -0.67 -2.22
1 1 5 2 1 0.66 4.29 0.68 1.55 4.20 1.54 0.60

5 Map C : Compressed Map without Novelty

Let us try to visualize the compressed Map C from the flow diagram below.

Figure 13:Flow map with highlighted bounding box in red around compressed map C

Figure 13:Flow map with highlighted bounding box in red around compressed map C

5.0.1 Iteration 1:

With the Novelties removed, we construct another hierarchical Voronoi tessellation map C layer 2 on the dataset without Novelty and below mentioned model parameters.

Model Parameters

map_C <- list()
mapA_scale_summary = map_A[[3]]$scale_summary
map_C <- muHVT::HVT(dataset_without_novelty,
                  n_cells = 10,
                  depth = 2,
                  quant.err = 0.2,
                  projection.scale = 10,
                  normalize = F,
                  distance_metric = "L1_Norm",
                  error_metric = "max",
                  quant_method = "kmeans",
                  diagnose = F,
                  scale_summary = mapA_scale_summary)

Now let’s check the compression summary for HVT (map C). The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

mapC_compression_summary <- map_C[[3]]$compression_summary %>%  dplyr::mutate_if(is.numeric, funs(round(.,4)))
compressionSummaryTable(mapC_compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 10 0 0 n_cells: 10 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans
2 100 7 0.07 n_cells: 10 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 0% of the cells have hit the quantization threshold error in level 1 and 7% of the cells have hit the quantization threshold error in level 2

5.0.2 Iteration 2:

Since, we are yet to achive atleast 80% compression at depth 2. Let’s try to compress again using the below mentioned set of model parameters.

Model Parameters

map_C <- list()
map_C <- muHVT::HVT(dataset_without_novelty,
                  n_cells = 23,    #23
                  depth = 2,
                  quant.err = 0.2,
                  projection.scale = 10,
                  normalize = F,
                  distance_metric = "L1_Norm",
                  error_metric = "max",
                  quant_method = "kmeans",
                  diagnose = F,
                  scale_summary = mapA_scale_summary)

The datatable displayed below is the summary from map C (layer2).

summaryTable(map_C[[3]]$summary,scroll = T,limit = 500)
Segment.Level Segment.Parent Segment.Child n Cell.ID Quant.Error price speed hd ram screen ads
1 1 1 391 512 0.56 -1.27 -1.04 -0.91 -0.83 -0.59 0.86
1 1 2 105 96 1.02 2.08 0.75 0.11 0.71 2.64 0.56
1 1 3 190 55 1.37 1.48 0.02 2.32 2.63 0.12 -0.22
1 1 4 86 15 0.94 0.87 1.27 2.61 2.57 1.33 -1.76
1 1 5 257 326 0.49 0.41 0.45 -0.08 -0.20 -0.67 0.66
1 1 6 116 201 0.77 1.57 0.45 -0.24 0.27 -0.30 -1.30
1 1 7 235 228 0.58 0.85 -0.82 0.42 1.37 -0.47 0.52
1 1 8 149 185 0.84 -0.79 1.73 0.76 -0.22 -0.28 -1.75
1 1 9 353 306 0.69 -0.77 0.55 0.28 -0.22 -0.17 -0.79
1 1 10 234 297 0.54 0.25 0.50 -0.11 -0.16 0.43 0.66
1 1 11 330 478 0.56 -1.18 -0.79 -0.70 -0.76 -0.52 -0.69
1 1 12 373 434 0.6 -0.50 0.40 -0.84 -0.84 -0.56 0.66
1 1 13 209 147 0.81 1.32 0.99 0.36 1.37 -0.01 0.70
1 1 14 337 471 0.36 -0.51 -0.97 -0.87 -0.75 -0.67 0.57
1 1 15 283 404 0.71 0.06 -0.40 -0.78 -0.40 -0.42 -1.42
1 1 16 168 142 0.85 0.13 0.66 0.48 -0.02 2.64 -0.97
1 1 17 285 384 0.53 -0.04 -0.96 0.05 -0.12 -0.65 0.54
1 1 18 118 301 0.75 0.35 2.30 -0.13 -0.29 -0.28 0.82
1 1 19 256 414 0.47 -0.47 -0.84 -0.43 -0.48 0.43 0.60
1 1 20 78 57 1 0.18 1.67 2.05 1.31 0.39 -1.97
1 1 21 171 360 0.64 0.24 -0.46 -0.37 -0.41 2.64 0.59
1 1 22 170 130 1.28 0.28 -0.01 1.64 0.99 0.21 -0.62
1 1 23 89 219 0.88 1.98 0.59 0.20 -0.20 0.12 0.58
2 1 1 19 528 0.09 -1.31 -1.17 -1.19 -1.11 -0.67 0.99
2 1 2 18 479 0.31 -1.42 -0.87 -0.21 -0.72 -0.67 0.53
2 1 3 28 521 0.1 -1.47 -1.22 -1.11 -0.76 -0.67 0.89
2 1 4 10 530 0.14 -1.84 -1.04 -1.20 -1.11 -0.67 0.79
2 1 5 27 519 0.1 -1.20 -1.09 -1.19 -1.11 -0.67 0.50
2 1 6 13 524 0.18 -1.36 -1.10 -0.99 -0.87 0.43 1.36
2 1 7 5 523 0.04 -1.19 -1.27 -0.78 -0.76 -0.67 1.57
2 1 8 23 494 0.06 -1.18 -0.89 -0.78 -0.76 -0.67 0.72
2 1 9 27 514 0.14 -1.35 -1.27 -1.08 -0.76 -0.67 0.39
2 1 10 35 510 0.1 -1.08 -1.27 -0.88 -0.76 -0.67 0.90
2 1 11 6 529 0.15 -1.22 -0.62 -1.19 -1.11 -0.67 1.50
2 1 12 10 513 0.07 -0.95 -0.89 -0.85 -0.76 -0.67 1.57
2 1 13 10 517 0.22 -1.70 -1.19 -0.73 -0.83 -0.67 0.57
2 1 14 11 472 0.17 -1.19 -1.13 -0.33 -0.05 -0.67 0.97
2 1 15 11 507 0.19 -1.00 -0.93 -1.17 -0.63 -0.67 0.97
2 1 16 7 531 0.07 -1.58 -1.27 -1.18 -0.96 -0.67 1.57
2 1 17 26 499 0.09 -1.31 -0.89 -0.92 -0.76 -0.67 0.36
2 1 18 23 515 0.07 -0.99 -0.89 -1.19 -1.11 -0.67 0.92
2 1 19 13 520 0.14 -1.56 -0.98 -0.74 -0.76 -0.67 1.34
2 1 20 15 489 0.12 -1.05 -0.89 -0.26 -0.76 -0.67 1.31
2 1 21 16 503 0.18 -1.48 -0.79 -0.73 -0.76 -0.67 0.76
2 1 22 15 518 0.26 -1.37 -1.07 -1.09 -1.07 0.43 0.66
2 1 23 23 500 0.07 -1.03 -0.89 -0.82 -0.76 -0.67 1.06
2 2 1 4 108 0.12 2.31 0.68 -0.12 -0.05 2.64 1.31
2 2 2 3 118 0.11 2.53 0.68 -0.47 -0.76 2.64 0.36
2 2 3 7 99 0.21 1.06 0.80 0.29 1.37 2.64 0.35
2 2 4 2 89 0.06 1.09 2.30 0.25 -0.05 2.64 1.57
2 2 5 7 59 0.13 3.21 0.68 -0.27 1.37 2.64 0.53
2 2 6 7 88 0.11 1.85 0.68 0.16 1.37 2.64 0.85
2 2 7 2 145 0.11 1.14 0.68 0.10 -0.05 2.64 1.36
2 2 8 3 38 0.11 3.32 0.68 -0.27 1.37 2.64 -0.78
2 2 9 6 100 0.17 2.86 0.68 0.10 -0.05 2.64 0.48
2 2 10 2 24 0.11 3.50 2.30 1.23 -0.05 2.64 -0.23
2 2 11 2 12 0.38 4.64 0.68 1.36 0.66 2.64 0.37
2 2 12 6 56 0.25 1.52 2.30 0.32 1.37 2.64 0.77
2 2 13 4 50 0.34 1.64 0.49 1.90 1.37 2.64 0.14
2 2 14 13 76 0.22 2.42 0.68 0.03 1.37 2.64 0.53
2 2 15 4 103 0.12 1.08 2.30 0.14 -0.05 2.64 0.52
2 2 16 11 167 0.13 1.23 0.68 -0.20 -0.05 2.64 0.59
2 2 17 1 107 0 1.99 -0.08 0.49 -0.05 2.64 -0.62
2 2 18 4 84 0.15 1.52 0.68 0.34 1.37 2.64 1.36
2 2 19 5 169 0.17 1.81 -0.08 -0.29 -0.19 2.64 0.99
2 2 20 4 70 0.11 2.69 -0.89 -0.27 1.37 2.64 0.69
2 2 21 4 122 0.11 1.99 0.68 0.22 -0.05 2.64 0.57
2 2 22 3 60 0.12 2.74 -0.89 -0.27 1.37 2.64 -0.78
2 2 23 1 195 0 1.63 0.68 -0.95 -0.76 2.64 0.69
2 3 1 5 39 0.04 1.54 -0.08 2.29 2.79 0.43 -0.81
2 3 2 3 9 0.05 1.71 -0.89 2.29 2.79 2.64 -0.55
2 3 3 10 51 0.07 1.10 -0.08 2.29 2.79 0.43 -0.56
2 3 4 5 32 0.39 3.74 -0.10 2.60 1.37 -0.67 0.96
2 3 5 3 19 0.02 2.16 2.30 2.29 2.79 -0.67 0.04
2 3 6 1 20 0 2.03 2.30 2.29 2.79 0.43 0.35
2 3 7 16 49 0.18 1.49 -0.89 2.29 2.79 0.43 -0.06
2 3 8 7 40 0.09 1.80 -0.08 2.29 2.79 0.43 -0.26
2 3 9 2 54 0.05 2.90 0.68 3.73 -0.05 -0.67 0.69
2 3 10 1 91 0 1.16 0.68 2.29 1.37 -0.67 0.04
2 3 11 7 46 0.05 1.16 0.68 2.29 2.79 0.43 -0.49
2 3 12 4 77 0.12 1.66 0.68 2.44 1.37 0.43 0.80
2 3 13 7 37 0.08 1.63 0.68 2.29 2.79 0.43 -0.73
2 3 14 12 53 0.1 1.69 0.68 2.29 2.79 -0.67 0.38
2 3 15 2 43 0.04 3.07 0.68 2.29 1.37 -0.67 -0.87
2 3 16 5 23 0.04 2.06 2.30 2.29 2.79 -0.67 0.35
2 3 17 12 45 0.06 0.92 0.68 2.29 2.79 0.43 -0.86
2 3 18 5 82 0.13 1.35 0.68 2.47 1.37 0.43 0.11
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2 15 9 14 369 0.19 0.28 -0.92 -0.38 -0.05 -0.67 -0.90
2 15 10 11 464 0.07 -0.06 -0.89 -0.82 -0.76 -0.67 -1.66
2 15 11 5 460 0.11 0.54 -0.89 -0.88 -0.76 -0.67 -1.70
2 15 12 7 410 0.17 0.51 -0.08 -0.92 -0.76 -0.67 -1.57
2 15 13 17 352 0.15 0.35 -0.89 -0.72 -0.13 0.43 -1.66
2 15 14 11 420 0.08 0.02 -1.03 -0.67 -0.05 -0.67 -1.65
2 15 15 13 396 0.13 0.43 -0.89 -0.71 -0.05 -0.67 -1.63
2 15 16 4 379 0.23 0.75 -0.69 -0.65 -0.76 0.15 -0.67
2 15 17 7 322 0.21 0.47 0.24 -0.72 -0.66 0.43 -1.52
2 15 18 25 343 0.22 0.36 -0.08 -0.63 -0.05 -0.67 -1.53
2 15 19 15 496 0.09 -0.38 -1.04 -1.11 -0.76 -0.67 -1.63
2 15 20 14 380 0.3 0.03 -0.08 -0.55 -0.46 -0.67 -0.82
2 15 21 14 293 0.21 0.50 -0.08 -0.67 -0.05 0.43 -1.14
2 15 22 10 353 0.17 0.20 0.68 -0.81 -0.48 -0.67 -0.92
2 15 23 11 432 0.09 -0.35 -0.89 -0.80 -0.05 -0.67 -1.61
2 16 1 8 58 0.25 0.04 2.30 1.70 -0.05 2.64 -1.20
2 16 2 1 33 0 1.25 2.30 1.20 1.37 2.64 -0.94
2 16 3 13 90 0.39 0.07 2.30 0.48 -0.05 2.64 -0.99
2 16 4 5 230 0.11 0.87 -0.89 -0.67 -0.33 2.64 -1.64
2 16 5 11 163 0.15 -0.13 0.68 0.38 -0.05 2.64 -0.78
2 16 6 4 75 0.04 0.84 -0.89 1.20 1.37 2.64 -0.83
2 16 7 17 207 0.2 -0.62 0.68 0.36 -0.76 2.64 -1.01
2 16 8 20 135 0.14 -0.24 0.98 0.51 -0.05 2.64 -1.10
2 16 9 9 132 0.15 1.33 -0.08 -0.65 -0.29 2.64 -1.52
2 16 10 8 74 0.2 0.95 0.39 1.10 1.37 2.64 -0.65
2 16 11 8 284 0.05 -0.76 -0.08 0.51 -0.76 2.64 -1.16
2 16 12 5 83 0.04 0.78 -0.89 1.20 1.37 2.64 -0.43
2 16 13 5 105 0.07 -0.44 0.68 0.49 -0.05 2.64 -2.23
2 16 14 4 136 0.09 1.24 0.68 -0.64 -0.05 2.64 -0.87
2 16 15 2 106 0.03 1.56 0.68 -0.71 -0.76 2.64 -1.71
2 16 16 9 120 0.06 -0.01 0.68 1.23 -0.05 2.64 -0.72
2 16 17 7 67 0.21 0.47 0.41 1.49 1.37 2.64 -0.97
2 16 18 7 213 0.33 0.07 0.57 -0.21 -0.05 2.64 -0.36
2 16 19 5 160 0.15 0.43 0.68 0.62 -0.05 2.64 0.17
2 16 20 6 320 0.13 -0.58 0.68 -0.26 -0.76 2.64 -0.44
2 16 21 3 85 0.06 0.38 0.68 2.29 -0.05 2.64 -0.61
2 16 22 6 345 0.1 -0.82 -0.08 0.21 -0.76 2.64 -0.79
2 16 23 5 110 0.06 1.39 0.68 -0.67 -0.05 2.64 -1.61
2 17 1 6 440 0.09 -0.58 -1.27 0.12 -0.05 -0.67 1.29
2 17 2 10 416 0.13 -0.58 -0.93 -0.18 -0.05 -0.67 0.64
2 17 3 12 417 0.09 -0.24 -0.92 0.15 -0.05 -0.67 1.57
2 17 4 9 349 0.12 0.29 -0.93 0.23 -0.05 -0.67 -0.11
2 17 5 5 309 0.28 0.52 -0.89 0.35 -0.05 0.43 0.52
2 17 6 16 385 0.13 0.31 -0.89 -0.45 -0.05 -0.67 0.41
2 17 7 23 372 0.09 0.31 -0.91 0.03 -0.05 -0.67 0.86
2 17 8 21 403 0.09 -0.15 -1.27 0.09 -0.05 -0.67 0.85
2 17 9 14 366 0.13 -0.37 -0.92 0.45 -0.05 -0.67 -0.31
2 17 10 11 419 0.13 -0.26 -0.89 -0.36 -0.05 -0.67 1.02
2 17 11 13 389 0.11 -0.35 -0.95 0.02 -0.05 -0.67 0.00
2 17 12 17 392 0.06 -0.26 -0.89 0.05 -0.05 -0.67 0.62
2 17 13 4 362 0.17 0.15 -1.08 0.01 -0.05 -0.67 -0.67
2 17 14 9 424 0.17 0.37 -0.89 -0.29 -0.76 -0.67 0.90
2 17 15 12 395 0.09 -0.06 -1.27 -0.09 -0.05 -0.67 0.37
2 17 16 18 399 0.11 -0.17 -0.89 0.09 -0.05 -0.67 1.05
2 17 17 15 373 0.09 -0.59 -0.89 0.45 -0.05 -0.67 -0.58
2 17 18 11 357 0.09 0.20 -0.89 0.47 -0.05 -0.67 0.68
2 17 19 15 398 0.26 0.98 -0.89 -0.41 -0.57 -0.67 0.49
2 17 20 20 367 0.08 0.25 -0.91 0.05 -0.05 -0.67 0.46
2 17 21 3 438 0.08 -0.60 -0.89 0.04 -0.76 -0.67 0.48
2 17 22 6 429 0.08 -0.22 -0.89 -0.07 -0.76 -0.67 0.68
2 17 23 15 382 0.08 -0.35 -0.92 0.45 -0.05 -0.67 0.47
2 18 1 4 430 0.06 -0.12 2.30 -0.78 -0.76 -0.67 0.52
2 18 2 5 199 0.07 0.53 2.30 0.46 -0.05 0.43 0.62
2 18 3 4 276 0.04 -0.21 2.30 0.03 -0.05 -0.67 -0.48
2 18 4 1 225 0 0.74 2.30 -0.19 -0.05 -0.67 -0.79
2 18 5 4 340 0.08 0.42 2.30 -0.56 -0.76 -0.67 -0.64
2 18 6 16 286 0.15 0.56 2.30 0.00 -0.05 -0.67 1.49
2 18 7 5 242 0.04 0.41 2.30 0.45 -0.05 -0.67 0.35
2 18 8 7 187 0.09 0.61 2.30 0.31 -0.05 0.43 1.57
2 18 9 8 200 0.15 0.62 2.30 0.08 -0.05 0.43 0.04
2 18 10 5 226 0.04 0.41 2.30 0.05 -0.05 0.43 0.69
2 18 11 8 463 0.07 0.24 2.30 -0.78 -0.76 -0.67 1.30
2 18 12 9 522 0.08 -0.14 2.30 -1.19 -1.11 -0.67 1.33
2 18 13 2 115 0.08 1.74 2.30 1.23 -0.05 -0.67 -0.03
2 18 14 6 274 0.18 0.31 2.30 -0.06 -0.05 -0.67 0.15
2 18 15 3 229 0.06 0.55 2.30 -0.06 -0.05 0.43 1.14
2 18 16 10 254 0.13 0.67 2.30 0.22 -0.05 -0.67 0.87
2 18 17 4 407 0.13 0.00 2.30 -0.78 -0.76 0.43 0.94
2 18 18 7 246 0.08 0.16 2.30 -0.04 -0.05 0.43 1.57
2 18 19 4 339 0.04 -0.04 2.30 0.05 -0.76 -0.67 0.04
2 18 20 3 217 0.01 0.65 2.30 0.03 -0.05 0.43 0.69
2 18 21 3 476 0.14 -0.52 2.30 -0.45 -0.76 -0.67 1.29
2 18 22 0 NA NA NA NA NA NA NA NA
2 18 23 0 NA NA NA NA NA NA NA NA
2 19 1 6 391 0.18 0.52 -0.89 -0.34 -0.64 0.43 0.98
2 19 2 6 383 0.07 0.68 -0.89 -0.56 -0.76 0.43 0.10
2 19 3 21 470 0.12 -1.02 -0.91 -0.76 -0.76 0.43 0.53
2 19 4 11 348 0.2 0.08 -1.03 -0.10 -0.05 0.43 0.14
2 19 5 10 446 0.15 -1.12 -0.81 -0.30 -0.76 0.43 0.01
2 19 6 14 350 0.08 0.07 -0.89 0.05 -0.05 0.43 0.74
2 19 7 8 388 0.27 -0.53 -0.38 -0.56 -0.40 0.43 -0.28
2 19 8 7 467 0.11 -1.18 -0.89 -0.29 -0.76 0.43 0.88
2 19 9 6 425 0.09 -0.49 -0.89 -0.23 -0.76 0.43 0.58
2 19 10 9 406 0.1 -0.18 -0.08 -0.75 -0.76 0.43 0.63
2 19 11 10 400 0.16 -0.36 -0.89 -0.71 -0.05 0.43 0.33
2 19 12 9 347 0.21 -0.22 -0.89 0.49 -0.13 0.43 0.63
2 19 13 8 497 0.1 -0.83 -0.89 -1.19 -1.11 0.43 0.44
2 19 14 16 378 0.12 -0.25 -1.03 -0.12 -0.05 0.43 0.74
2 19 15 16 418 0.15 -0.73 -0.08 -0.41 -0.76 0.43 0.82
2 19 16 22 448 0.15 -0.21 -0.94 -0.82 -0.76 0.43 0.77
2 19 17 13 485 0.15 -0.74 -0.95 -0.83 -0.76 0.43 1.20
2 19 18 13 447 0.14 -0.52 -0.98 -0.75 -0.76 0.43 0.17
2 19 19 10 412 0.21 -0.48 -0.81 -0.43 -0.05 0.43 1.25
2 19 20 11 355 0.16 -0.61 -0.89 0.20 -0.05 0.43 -0.14
2 19 21 9 390 0.1 -0.66 -0.89 -0.14 -0.05 0.43 0.65
2 19 22 10 401 0.15 -0.52 -0.93 0.13 -0.05 0.43 1.40
2 19 23 11 487 0.12 -0.83 -1.27 -0.86 -0.76 0.43 0.75
2 20 1 5 73 0.02 -0.13 0.68 1.70 1.37 0.43 -2.46
2 20 2 3 69 0.02 -0.02 0.68 2.29 1.37 0.43 -1.98
2 20 3 3 65 0.19 -0.83 1.22 3.07 -0.05 -0.67 -2.46
2 20 4 7 79 0.03 0.82 2.30 1.21 1.37 0.43 -0.92
2 20 5 5 62 0.03 0.13 0.68 2.29 1.37 0.43 -2.32
2 20 6 3 80 0.04 -0.03 0.68 1.70 1.37 0.43 -2.19
2 20 7 3 71 0.03 0.47 2.30 1.70 1.37 0.43 -0.79
2 20 8 2 72 0 0.22 0.68 1.70 1.37 0.43 -2.46
2 20 9 12 35 0.14 0.15 2.30 2.29 1.37 0.43 -2.29
2 20 10 4 44 0.06 -0.03 0.68 3.07 1.37 0.43 -2.25
2 20 11 11 42 0.07 0.15 2.30 1.70 1.37 0.43 -2.39
2 20 12 1 31 0 0.57 2.30 3.30 1.37 0.43 -1.25
2 20 13 5 63 0.08 -0.29 0.68 2.29 1.37 0.43 -2.35
2 20 14 6 66 0.05 0.39 2.30 1.70 1.37 0.43 -1.23
2 20 15 1 68 0 0.31 0.68 2.29 1.37 0.43 -1.98
2 20 16 3 78 0.05 1.06 2.30 1.23 1.37 0.43 -0.62
2 20 17 4 27 0.05 0.19 2.30 3.07 1.37 0.43 -2.25
2 20 18 0 NA NA NA NA NA NA NA NA
2 20 19 0 NA NA NA NA NA NA NA NA
2 20 20 0 NA NA NA NA NA NA NA NA
2 20 21 0 NA NA NA NA NA NA NA NA
2 20 22 0 NA NA NA NA NA NA NA NA
2 20 23 0 NA NA NA NA NA NA NA NA
2 21 1 4 504 0.17 0.72 -0.89 -0.66 -0.76 2.64 1.36
2 21 2 5 245 0.17 0.97 0.68 -0.65 -0.62 2.64 1.03
2 21 3 6 313 0.16 -0.20 0.68 -0.40 -0.52 2.64 0.41
2 21 4 10 338 0.18 0.27 -0.89 -0.34 -0.05 2.64 0.32
2 21 5 4 375 0.13 0.16 -0.49 0.04 -0.76 2.64 0.66
2 21 6 6 298 0.15 1.35 -0.08 -0.45 -0.76 2.64 0.94
2 21 7 10 526 0.21 -0.31 -1.08 -0.88 -0.76 2.64 1.28
2 21 8 11 490 0.15 -0.83 -0.89 -0.25 -0.76 2.64 -0.33
2 21 9 6 393 0.19 1.39 -0.89 -0.56 -0.76 2.64 -0.04
2 21 10 8 359 0.11 0.49 -0.89 -0.39 -0.05 2.64 0.95
2 21 11 6 295 0.1 0.19 -0.08 -0.29 -0.05 2.64 0.75
2 21 12 7 376 0.09 -0.05 -0.89 -0.19 -0.05 2.64 0.77
2 21 13 18 205 0.2 0.58 0.68 -0.13 -0.13 2.64 0.65
2 21 14 6 186 0.18 0.55 0.55 0.01 -0.05 2.64 1.50
2 21 15 6 328 0.14 0.27 -0.89 -0.73 -0.05 2.64 -0.76
2 21 16 2 215 0.1 0.75 -0.08 -0.46 -0.05 2.64 -0.35
2 21 17 5 408 0.12 -0.53 -0.08 -0.29 -0.76 2.64 0.19

Now let’s check the compression summary for HVT (map C). The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

mapC_compression_summary <- map_C[[3]]$compression_summary %>%  dplyr::mutate_if(is.numeric, funs(round(.,4)))
compressionSummaryTable(mapC_compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 23 0 0 n_cells: 23 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans
2 508 434 0.85 n_cells: 23 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 0% of the cells have hit the quantization threshold error in level 1 and 85% of the cells have hit the quantization threshold error in level 2

Let’s plot the Voronoi tessellation for layer 2 (map C)

muHVT::plotHVT(map_C,
        line.width = c(0.4,0.2), 
        color.vec = c("#141B41","#0582CA"),
        centroid.size = 0.1,
        maxDepth = 2) 
Figure 14: The Voronoi Tessellation for layer 2 (map C) shown for the 100 cells in the dataset ’computers’ at level 2

Figure 14: The Voronoi Tessellation for layer 2 (map C) shown for the 100 cells in the dataset ’computers’ at level 2

Heat Maps

Now let’s plot all the features for each cell at level two as a heatmap for better visualization.

The heatmaps displayed below provides a visual representation of the spatial characteristics of the computers data, allowing us to observe patterns and trends in the distribution of each of the features (n,price,speed,hd,ram,screen,ads). The sheer green shades highlight regions with higher values in each of the heatmaps, while the indigo shades indicate areas with the lowest values in each of the heatmaps. By analyzing these heatmaps, we can gain insights into the variations and relationships between each of these features within the computers data.


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "n",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
) 
Figure 15: The Voronoi Tessellation with the heat map overlaid for features No. of entities in each cell

Figure 15: The Voronoi Tessellation with the heat map overlaid for features No. of entities in each cell


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "price",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
) 
Figure 16: The Voronoi Tessellation with the heat map overlaid for features price in the ’computers’ dataset

Figure 16: The Voronoi Tessellation with the heat map overlaid for features price in the ’computers’ dataset


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "speed",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
) 
Figure 17: The Voronoi Tessellation with the heat map overlaid for features speed in the ’computers’ dataset

Figure 17: The Voronoi Tessellation with the heat map overlaid for features speed in the ’computers’ dataset


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "hd",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
) 
Figure 18: The Voronoi Tessellation with the heat map overlaid for features hd in the ’computers’ dataset

Figure 18: The Voronoi Tessellation with the heat map overlaid for features hd in the ’computers’ dataset


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "ram",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
) 
Figure 19: The Voronoi Tessellation with the heat map overlaid for features ram in the ’computers’ dataset

Figure 19: The Voronoi Tessellation with the heat map overlaid for features ram in the ’computers’ dataset


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "screen",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
) 
Figure 20: The Voronoi Tessellation with the heat map overlaid for features screen in the ’computers’ dataset

Figure 20: The Voronoi Tessellation with the heat map overlaid for features screen in the ’computers’ dataset


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "ads",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
) 
Figure 21: The Voronoi Tessellation with the heat map overlaid for features ads in the ’computers’ dataset

Figure 21: The Voronoi Tessellation with the heat map overlaid for features ads in the ’computers’ dataset

We now have the set of maps (map A, map B & map C) which will be used to predict which map and cell each test record is assigned to, but before that lets view our test dataset

6 Prediction on Test Data

Now once we have built the model, let us try to predict using our test dataset which cell and which layer each point belongs to.

Raw Testing Dataset

The testing dataset includes the following columns:

Let’s have a look at our randomly selected test dataset containing 1253 datapoints.

Table(head(testComputers_data))
Row.No price speed hd ram screen ads
3 1595 25 170 4 15 94
4 1849 25 170 8 14 94
7 1720 25 170 4 14 94
10 2575 50 210 4 15 94
11 2195 33 170 8 15 94
14 2295 25 245 8 14 94

The predictLayerHVT function is used to score the test data using the predictive set of maps. This function takes an input - a test data and a set of maps (map A, map B, map C).

Now, Let us understand the predictLayerHVT function.

predictLayerHVT(data,
                map_A,
                map_B,
                map_C,
                mad.threshold = 0.2,
                normalize = T, 
                distance_metric="L1_Norm",
                error_metric="max",
                child.level = 1, 
                line.width = c(0.6, 0.4, 0.2),
                color.vec = c("#141B41", "#6369D1", "#D8D2E1"),
                yVar= NULL,
                ...)

Each of the parameters of predictLayerHVT function has been explained below:

The function predicts based on the HVT maps - map A, map B and map C, constructed using HVT function. For each test record, the function will assign that record to Layer1 or Layer2. Layer1 contains the cell ids from map A and Layer 2 contains cell ids from map B (novelty map) and map C (map without novelty).

Prediction Algorithm

The prediction algorithm recursively calculates the distance between each point in the test dataset and the cell centroids for each level. The following steps explain the prediction method for a single point in the test dataset:

  1. Calculate the distance between the point and the centroid of all the cells in the first level.
  2. Find the cell whose centroid has minimum distance to the point.
  3. Check if the cell drills down further to form more cells.
  4. If it doesn’t, return the path. Or else repeat steps 1 to 4 till we reach a level at which the cell doesn’t drill down further.

Note : The prediction algorithm will not work if some of the variables used to perform quantization are missing. In the test dataset, we should not remove any features

Let’s see which cell and layer each point belongs to and check the Mean Absolute Difference.


validation_data <- testComputers
new_predict <- predictLayerHVT(
    data=validation_data,
    map_A,
    map_B,
    map_C,
    normalize = T
  )
summary_list <- map_A[[3]]

  train_colnames <- names(summary_list[["nodes.clust"]][[1]][[1]])
  scaled_test_data <- scale(
     testComputers[, train_colnames],
      center = summary_list$scale_summary$mean_data[train_colnames],
      scale = summary_list$scale_summary$std_data[train_colnames])
testComputers <- scaled_test_data
data1 <- data.frame(testComputers)
data1$Row.No <- row.names(testComputers_data)
data1 <- data1 %>% dplyr::select(Row.No,price,speed,hd,ram,screen,ads)
colnames(data1) <- c("Row.No","price_act","speed_act","hd_act","ram_act","screen_act","ads_act")
Layer2.Cell.ID <- new_predict$Layer2.Cell.ID
combined <- cbind(data1,Layer2.Cell.ID)
combined <- combined %>%mutate(Cell.ID = gsub("[BC]", "", Layer2.Cell.ID))
mapC_summary <- map_C[[3]]$summary

df1 <- combined
df1$Cell.ID <- as.numeric(df1$Cell.ID)
df2 <- mapC_summary
merged_df <- merge(df1, df2, by = "Cell.ID", all.x = TRUE)
sorted_df <- merged_df[order(merged_df$Row.No), ]
sorted_df <- sorted_df %>% select(price_act,speed_act,hd_act,ram_act,screen_act,ads_act,Layer2.Cell.ID,price,speed,hd,ram,screen,ads)
sorted_df$Row.No <- testComputers_data$Row.No
sorted_df <- sorted_df%>% dplyr::select(Row.No,price_act,speed_act,hd_act,ram_act,screen_act,ads_act,Layer2.Cell.ID,price,speed,hd,ram,screen,ads)
colnames(sorted_df) <- c("Row.No","price_act","speed_act","hd_act","ram_act","screen_act","ads_act","Layer2.Cell.ID","price_pred","speed_pred","hd_pred","ram_pred","screen_pred","ads_pred")
sorted_df$diff <- rowMeans(abs(sorted_df[, c("price_act","speed_act","hd_act","ram_act","screen_act","ads_act")] - sorted_df[, c("price_pred","speed_pred","hd_pred","ram_pred","screen_pred","ads_pred")]))
rownames(sorted_df) <- NULL
options(scipen = 999)
sorted_df %>% head(1000) %>%as.data.frame() %>%Table(scroll = T)
Row.No price_act speed_act hd_act ram_act screen_act ads_act Layer2.Cell.ID price_pred speed_pred hd_pred ram_pred screen_pred ads_pred diff
3 -1.0785679 -1.2710382 -0.9472574 -0.7589986 0.4307274 -1.7213059 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.4690860
4 1.1382448 -0.0817113 -0.7914378 -0.7589986 2.6404120 -1.7213059 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.7270993
7 1.3374115 -0.0817113 0.1512706 1.3676416 0.4307274 -1.1163339 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.5289052
10 -1.0629809 0.6794579 -0.7758559 -0.7589986 -0.6741149 -0.5382496 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3296222
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1097 -0.7702924 0.6794579 1.7016753 -0.0501185 -0.6741149 -1.9767386 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.4662247
1112 -0.3875458 2.2969425 1.7016753 -0.0501185 -0.6741149 -1.9767386 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.4500887
1115 0.0471573 2.2969425 2.2859986 1.3676416 0.4307274 -1.9767386 C57 0.1798241 1.6748331 2.0467757 1.3131124 0.3882335 -1.9684654 0.1832159
1125 0.0471573 -0.8904536 -0.6356183 -0.0501185 0.4307274 -0.6189125 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.4054435
1130 -1.0439302 2.2969425 2.2859986 -0.7589986 -0.6741149 -1.9767386 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.5849315
1138 0.4801285 -0.0817113 3.0650965 2.7854017 0.4307274 -1.9767386 C15 0.8693193 1.2680365 2.6053382 2.5710892 1.3300177 -1.7644512 0.5874312
1141 -0.8187852 2.2969425 1.7016753 -0.0501185 -0.6741149 -1.9767386 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.3873786
1145 -0.6213503 0.6794579 0.4940736 -0.0501185 2.6404120 -1.9767386 C142 0.1328959 0.6627507 0.4814828 -0.0248014 2.6404120 -0.9712527 0.3023912
1155 -1.1668940 -0.0817113 0.4473277 -0.7589986 -0.6741149 -1.9767386 B1 1.0281464 1.1551887 3.0650965 2.7854017 2.6404120 -2.3416423 2.2122567
1160 -0.7252634 0.6794579 1.2342166 -0.0501185 -0.6741149 -1.9767386 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.3958197
1172 -0.0480964 -0.0817113 3.0650965 1.3676416 0.4307274 -1.9767386 C57 0.1798241 1.6748331 2.0467757 1.3131124 0.3882335 -1.9684654 0.5180137
1174 0.6689040 1.1076156 3.0650965 2.7854017 0.4307274 -1.9767386 C15 0.8693193 1.2680365 2.6053382 2.5710892 1.3300177 -1.7644512 0.3577474
1177 0.0090558 -0.0817113 2.2859986 1.3676416 0.4307274 -1.9767386 C57 0.1798241 1.6748331 2.0467757 1.3131124 0.3882335 -1.9684654 0.3786387
1187 -0.5971039 2.2969425 1.7016753 -0.0501185 -0.6741149 -1.9767386 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.4151624
1188 1.5192595 -0.8904536 -0.2850242 1.3676416 -0.6741149 -1.7213059 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.5546423
1196 1.1642231 -0.8904536 0.1512706 1.3676416 0.4307274 -0.6189125 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.4470960
1197 -1.0716404 0.6794579 0.4940736 -0.0501185 -0.6741149 -1.9767386 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.3960171
1200 -1.0785679 -0.0817113 1.7016753 -0.0501185 -0.6741149 -1.9767386 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.6374643
1206 -1.3470101 0.6794579 0.4473277 -0.7589986 0.4307274 -1.9767386 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.5344366
1218 -0.9140388 -0.0817113 1.7016753 -0.0501185 0.4307274 -1.9767386 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.6596138
1220 -1.3383506 2.2969425 0.4473277 -0.7589986 -0.6741149 -1.9767386 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.4309743
1221 0.4783966 2.2969425 2.2859986 1.3676416 0.4307274 -1.9767386 C57 0.1798241 1.6748331 2.0467757 1.3131124 0.3882335 -1.9684654 0.2108669
1228 -1.3383506 2.2969425 0.4473277 -0.7589986 -0.6741149 -2.2859465 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.4825090
1242 -1.4959522 0.6794579 0.4473277 -0.7589986 -0.6741149 -2.2859465 B1 1.0281464 1.1551887 3.0650965 2.7854017 2.6404120 -2.3416423 2.0887035
1247 -1.1738216 2.2969425 0.4473277 -0.7589986 0.4307274 -2.2859465 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.5082286
1253 -0.8984519 0.6794579 0.4940736 -0.0501185 -0.6741149 -2.2859465 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.4186870
1254 0.1337515 0.6794579 -0.7758559 -0.7589986 -0.6741149 -0.6189125 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.4101111
1259 -0.4724082 2.2969425 0.4940736 -0.0501185 0.4307274 -2.2859465 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.4271898
1269 -1.0439302 2.2969425 2.2859986 -0.7589986 -0.6741149 -2.2859465 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.6364662
1270 -1.3314231 0.6794579 0.4940736 -0.0501185 -0.6741149 -2.2859465 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.4908489
1277 0.9113679 1.1076156 3.0650965 2.7854017 0.4307274 -2.2859465 C15 0.8693193 1.2680365 2.6053382 2.5710892 1.3300177 -1.7644512 0.3828876
1282 -0.9140388 0.6794579 1.7016753 -0.0501185 0.4307274 -2.2859465 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.5878571
1283 -0.6473286 0.6794579 1.7016753 -0.0501185 -0.6741149 -2.2993903 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.5404939
1285 -1.5981334 0.6794579 0.4473277 -0.7589986 -0.6741149 -2.2993903 B1 1.0281464 1.1551887 3.0650965 2.7854017 2.6404120 -2.3416423 2.1034931
1287 -0.9850461 0.6794579 0.4940736 -0.0501185 -0.6741149 -2.2993903 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.4353600
1288 -0.9140388 0.6794579 1.7016753 -0.0501185 0.4307274 -2.2993903 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.5900977
1298 -0.1104442 0.6794579 2.2859986 1.3676416 0.4307274 -2.2993903 C57 0.1798241 1.6748331 2.0467757 1.3131124 0.3882335 -1.9684654 0.3254691
1302 -1.2084593 -1.2710382 -1.2978514 -0.7589986 -0.6741149 -0.6189125 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.2219506
1303 -1.3383506 0.6794579 2.2859986 -0.7589986 -0.6741149 -2.2993903 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.7677039
1305 0.1493385 2.2969425 2.2859986 1.3676416 0.4307274 -2.2993903 C57 0.1798241 1.6748331 2.0467757 1.3131124 0.3882335 -1.9684654 0.2199610
1310 -0.3875458 2.2969425 1.7016753 -0.0501185 -0.6741149 -2.2993903 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.5038640
1311 0.2688386 2.2969425 2.2859986 1.3676416 0.4307274 -2.2993903 C57 0.1798241 1.6748331 2.0467757 1.3131124 0.3882335 -1.9684654 0.2297158
1320 0.5580633 2.2969425 2.2859986 1.3676416 2.6404120 -2.2993903 C57 0.1798241 1.6748331 2.0467757 1.3131124 0.3882335 -1.9684654 0.6462007
1329 -0.5971039 2.2969425 1.7016753 -0.0501185 -0.6741149 -2.2993903 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.4689377
1336 -0.9937055 2.2969425 0.4473277 -0.7589986 -0.6741149 -2.4472723 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.4519557
1348 -0.3009516 2.2969425 1.7016753 -0.0501185 0.4307274 -2.4472723 C57 0.1798241 1.6748331 2.0467757 1.3131124 0.3882335 -1.9684654 0.5554195
1367 -1.4336043 0.6794579 0.4473277 -0.7589986 0.4307274 -2.4472723 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.6272913
1375 -0.3944734 0.6794579 1.7016753 -0.0501185 2.6404120 -2.4472723 C142 0.1328959 0.6627507 0.4814828 -0.0248014 2.6404120 -0.9712527 0.5442676
1377 0.9130997 -0.8904536 0.4940736 -0.0501185 -0.6741149 -0.6189125 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.4510996
1382 -0.1104442 0.6794579 2.2859986 1.3676416 0.4307274 -2.4472723 C57 0.1798241 1.6748331 2.0467757 1.3131124 0.3882335 -1.9684654 0.3501161
1383 -1.5981334 0.6794579 0.4473277 -0.7589986 -0.6741149 -2.4472723 B1 1.0281464 1.1551887 3.0650965 2.7854017 2.6404120 -2.3416423 2.1140561
1387 0.9113679 1.1076156 3.0650965 2.7854017 0.4307274 -2.4472723 C15 0.8693193 1.2680365 2.6053382 2.5710892 1.3300177 -1.7644512 0.4097753
1388 1.5677522 1.1076156 4.6232922 4.2031619 0.4307274 -2.4607161 B1 1.0281464 1.1551887 3.0650965 2.7854017 2.6404120 -2.3416423 0.9819822
1390 -0.8984519 0.6794579 0.8446676 -0.0501185 0.4307274 -2.4607161 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.4715529
1393 -0.6542561 2.2969425 3.0650965 -0.0501185 -0.6741149 -2.4607161 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.7135368
1396 0.0090558 0.6794579 1.7016753 1.3676416 0.4307274 -2.4607161 C57 0.1798241 1.6748331 2.0467757 1.3131124 0.3882335 -1.9684654 0.3500863
1413 1.5192595 2.2969425 4.6232922 4.2031619 0.4307274 -2.4607161 B1 1.0281464 1.1551887 3.0650965 2.7854017 2.6404120 -2.3416423 1.1562635
1414 1.7773103 1.1076156 4.6232922 4.2031619 0.4307274 -2.4607161 B1 1.0281464 1.1551887 3.0650965 2.7854017 2.6404120 -2.3416423 1.0169086
1434 0.7018098 0.6794579 3.0650965 2.7854017 0.4307274 -2.4607161 C15 0.8693193 1.2680365 2.6053382 2.5710892 1.3300177 -1.7644512 0.5042857
1440 0.1250921 -0.0817113 -0.7758559 -0.7589986 0.4307274 -0.6189125 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3977861
1441 0.9113679 0.6794579 3.0650965 2.7854017 0.4307274 -2.4607161 C15 0.8693193 1.2680365 2.6053382 2.5710892 1.3300177 -1.7644512 0.4833755
1443 0.9823751 0.6794579 4.6232922 2.7854017 0.4307274 -2.4607161 C15 0.8693193 1.2680365 2.6053382 2.5710892 1.3300177 -1.7644512 0.7549094
1451 0.6533170 2.2969425 3.0650965 2.7854017 0.4307274 -2.4607161 C15 0.8693193 1.2680365 2.6053382 2.5710892 1.3300177 -1.7644512 0.5857557
1453 1.5677522 1.1076156 4.6232922 4.2031619 0.4307274 -2.4607161 B1 1.0281464 1.1551887 3.0650965 2.7854017 2.6404120 -2.3416423 0.9819822
1454 1.1642231 2.2969425 3.0650965 2.7854017 2.6404120 -2.4607161 C15 0.8693193 1.2680365 2.6053382 2.5710892 1.3300177 -1.7644512 0.6674233
1455 0.2982806 0.6794579 1.7016753 1.3676416 2.6404120 -2.4607161 C57 0.1798241 1.6748331 2.0467757 1.3131124 0.3882335 -1.9684654 0.7096484
1459 -0.6473286 2.2969425 0.4473277 -0.0501185 -0.6741149 -2.4607161 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.3818145
1461 1.0862882 2.2969425 4.6232922 2.7854017 0.4307274 -2.4607161 C15 0.8693193 1.2680365 2.6053382 2.5710892 1.3300177 -1.7644512 0.8456161
1480 -1.0785679 2.2969425 0.4473277 -0.0501185 -0.6741149 -2.4607161 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.4056986
1483 0.8749983 2.2969425 3.0650965 2.7854017 0.4307274 -2.4607161 C15 0.8693193 1.2680365 2.6053382 2.5710892 1.3300177 -1.7644512 0.5507019
1486 -0.3338574 0.6794579 -0.9472574 -0.7589986 -0.6741149 -0.6189125 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.3383311
1491 -1.1495752 0.6794579 0.4473277 -0.0501185 -0.6741149 -2.4607161 C185 -0.7912959 1.7279813 0.7575812 -0.2166340 -0.2811173 -1.7525245 0.4974602
1496 1.3460710 0.6794579 4.6232922 4.2031619 0.4307274 -2.4607161 B1 1.0281464 1.1551887 3.0650965 2.7854017 2.6404120 -2.3416423 1.0163949
1497 0.7554982 2.2969425 3.0650965 2.7854017 0.4307274 -2.4607161 C15 0.8693193 1.2680365 2.6053382 2.5710892 1.3300177 -1.7644512 0.5687255
1500 0.3069400 0.6794579 -0.6356183 -0.7589986 -0.6741149 -0.6189125 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.4545456
1502 -0.5520749 -0.8904536 -0.6356183 -0.7589986 -0.6741149 -0.6189125 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.1704858
1504 -0.5538068 -0.8904536 -1.1030770 -0.7589986 -0.6741149 -0.6189125 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.2266502
1507 1.3374115 -0.0817113 0.1512706 1.3676416 0.4307274 -0.6189125 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.5091823
1508 0.6533170 -0.8904536 -0.7914378 -0.0501185 2.6404120 -1.7213059 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.6569287
1510 -0.5070459 -0.0817113 -0.9472574 -0.7589986 0.4307274 -0.6189125 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.4444599
1511 -1.0785679 -0.8904536 -0.9472574 -0.7589986 0.4307274 -0.6189125 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.2471404
1514 0.9650563 -0.0817113 -0.6356183 -0.0501185 2.6404120 -0.6189125 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.4906051
1515 0.1250921 -0.0817113 -0.7758559 -0.7589986 -0.6741149 -0.6189125 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2995345
1519 0.6602445 -0.0817113 0.1434796 -0.0501185 -0.6741149 -0.6189125 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.4054230
1525 0.4108531 -0.8904536 -0.6356183 -0.0501185 2.6404120 -0.6189125 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.4068159
1527 -0.7321909 0.6794579 -1.1926732 -1.1134387 -0.6741149 -0.6189125 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.4209170
1528 2.3332454 0.6794579 0.4940736 -0.0501185 2.6404120 -0.6189125 C142 0.1328959 0.6627507 0.4814828 -0.0248014 2.6404120 -0.9712527 0.4345508
1530 0.2203458 -0.0817113 -0.6356183 -0.0501185 0.4307274 -0.6189125 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.4213515
1538 -1.4336043 -1.2710382 -1.1926732 -1.1134387 -0.6741149 -0.6189125 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3010184
1539 0.3242589 -0.0817113 -0.7914378 -0.0501185 -0.6741149 -1.7213059 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2493188
1555 -1.1062781 -1.2710382 -1.1420319 -0.7589986 -0.6741149 -0.6189125 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.2044949
1558 -0.2056979 -0.8904536 -1.1420319 -0.7589986 -0.6741149 -0.6189125 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.2911609
1563 1.5106000 0.6794579 0.1512706 1.3676416 0.4307274 -0.6189125 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.4111855
1570 -0.7321909 -1.2710382 -0.7758559 -0.7589986 -0.6741149 -0.6189125 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.2058135
1576 1.3529985 0.6794579 -0.2655468 1.3676416 0.4307274 0.3624865 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.2908216
1577 -0.5676619 -1.2710382 -0.7758559 -0.7589986 0.4307274 0.3624865 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.2319777
1581 -0.0325094 0.6794579 -0.2850242 -0.7589986 -0.6741149 0.3624865 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.2891217
1586 -0.2212849 0.6794579 -1.1926732 -1.1134387 -0.6741149 0.3624865 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.2649954
1590 0.9113679 -0.8904536 0.1512706 1.3676416 -0.6741149 0.3624865 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.1262013
1594 -0.5520749 -0.8904536 -0.7135281 -0.7589986 -0.6741149 0.3624865 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.0803848
1596 0.3588966 -0.0817113 -0.7914378 -0.0501185 -0.6741149 -1.7213059 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2550918
1602 -0.5867126 -0.0817113 -0.7914378 -0.7589986 0.4307274 0.3624865 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.2907739
1603 -0.5676619 -0.8904536 -1.1926732 -1.1134387 -0.6741149 0.3624865 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1709408
1617 -0.8620823 -0.8904536 -0.7836469 -0.7589986 -0.6741149 0.3624865 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1203663
1624 -0.9140388 -0.8904536 -1.1926732 -1.1134387 -0.6741149 0.3624865 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.2286703
1626 -0.8118576 -0.8904536 -0.9472574 -0.7589986 -0.6741149 0.3624865 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1116641
1633 -1.0872273 -1.2710382 -1.1926732 -1.1134387 -0.6741149 0.3624865 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.2591844
1642 -0.1277631 -0.0817113 -1.1926732 -1.1134387 -0.6741149 0.3624865 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.3128929
1645 1.0066215 0.6794579 0.1434796 -0.0501185 0.4307274 0.3624865 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.2650227
1649 0.8334330 -0.8904536 -0.2655468 1.3676416 0.4307274 0.3624865 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.3048784
1659 0.7381794 -0.8904536 -0.2850242 -0.0501185 -0.6741149 0.3624865 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.2403967
1669 0.4870561 -0.8904536 -0.7836469 -0.0501185 -0.6741149 -1.7213059 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3049914
1670 1.0066215 -0.8904536 -0.6745732 -0.7589986 -0.6741149 0.3624865 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.3336032
1671 0.3138676 -0.0817113 -0.2850242 -0.0501185 -0.6741149 0.3624865 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.2137875
1681 0.3069400 -0.0817113 -0.2850242 -0.0501185 0.4307274 0.3624865 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.2052761
1683 -0.7321909 -0.8904536 -1.1926732 -1.1134387 -0.6741149 0.3624865 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1983623
1688 -0.2126255 -0.8904536 -0.7758559 -0.7589986 -0.6741149 0.3624865 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1135150
1692 0.0125196 0.6794579 -0.6356183 -0.7589986 -0.6741149 0.3624865 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.2497122
1695 -0.7321909 -1.2710382 -0.7758559 -0.7589986 -0.6741149 0.3624865 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1359666
1697 2.6016875 0.6794579 0.3382540 -0.0501185 -0.6741149 0.3624865 C219 1.9755430 0.5864499 0.2031375 -0.2014525 0.1203785 0.5809108 0.3364201
1704 0.6533170 0.6794579 -0.2850242 -0.0501185 -0.6741149 0.3624865 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.1899706
1728 0.6966141 -0.0817113 -0.2850242 -0.0501185 0.4307274 0.3624865 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.2702218
1729 0.8265055 -0.8904536 -0.6550957 -0.0501185 -0.6741149 -1.7213059 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3822895
1731 0.3918024 -0.8904536 -0.2850242 -0.0501185 -0.6741149 0.3624865 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.1826672
1742 1.5106000 0.6794579 0.1512706 1.3676416 -0.6741149 0.3624865 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.2855043
1745 -0.2143573 -0.8904536 -0.7758559 -0.7589986 -0.6741149 0.3624865 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1132263
1756 0.7918678 0.6794579 -0.2850242 -0.0501185 2.6404120 0.3624865 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.3964077
1759 1.3529985 0.6794579 -0.3629340 -0.0501185 0.4307274 0.3624865 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.3240230
1771 1.3460710 0.6794579 0.1512706 1.3676416 -0.6741149 0.3624865 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.2580828
1772 0.8178461 0.6794579 -0.2850242 -0.0501185 -0.6741149 0.3624865 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.2173921
1774 0.9113679 0.6794579 -0.2850242 -0.0501185 -0.6741149 0.3624865 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.2329790
1784 -0.3875458 -1.2710382 -0.7758559 -0.7589986 -0.6741149 0.3624865 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1203119
1798 -1.2517564 -0.8904536 -0.7836469 -0.7589986 -0.6741149 0.3624865 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.1570941
1800 -0.8118576 -1.2710382 -1.1420319 -0.7589986 -0.6741149 -1.7213059 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.4120799
1811 -0.2992197 -0.8904536 -0.6356183 -0.0501185 0.4307274 0.3624865 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.1841475
1813 -0.0411688 0.6794579 -1.1926732 -1.1134387 -0.6741149 0.3624865 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.2950147
1818 2.4094483 0.6794579 0.4434322 -0.7589986 0.4307274 0.3624865 C219 1.9755430 0.5864499 0.2031375 -0.2014525 0.1203785 0.5809108 0.3089213
1819 -0.8187852 -0.8904536 -0.7914378 -0.7589986 -0.6741149 0.3624865 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1118516
1824 2.3852019 0.6794579 0.3382540 -0.0501185 0.4307274 0.3624865 C219 1.9755430 0.5864499 0.2031375 -0.2014525 0.1203785 0.5809108 0.2196485
1826 1.6041218 -0.0817113 0.1512706 1.3676416 -0.6741149 0.3624865 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.3539800
1832 -1.2517564 -0.8904536 -0.7836469 -0.7589986 -0.6741149 1.0212337 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.1014105
1845 0.2186139 -0.8904536 0.0499878 -0.0501185 -0.6741149 1.0212337 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.1496275
1858 0.9979621 -0.8904536 0.4940736 1.3676416 -0.6741149 1.0212337 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.1655488
1865 2.2189410 0.6794579 0.1434796 -0.0501185 2.6404120 1.0212337 C96 2.0782294 0.7478725 0.1122044 0.7127714 2.6404120 0.5611990 0.2438877
1871 1.3460710 0.6794579 -0.2850242 -0.0501185 0.4307274 -1.7213059 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.3298528
1872 0.8265055 0.6794579 -0.2850242 -0.7589986 -0.6741149 1.0212337 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.2961914
1875 -0.9053794 -1.2710382 -0.7758559 -0.7589986 -0.6741149 1.0212337 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.1741061
1876 -0.7321909 -0.0817113 -1.1926732 -1.1134387 -0.6741149 1.0212337 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.2995067
1877 -1.0716404 -0.0817113 -0.9472574 -0.7589986 -0.6741149 1.0212337 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.2514187
1889 3.9421665 -0.8904536 2.6755476 1.3676416 -0.6741149 1.0212337 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 1.0198285
1890 0.3918024 -0.8904536 0.0499878 -0.0501185 -0.6741149 1.0212337 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.1784923
1892 0.6429257 0.6794579 -0.2850242 -0.0501185 2.6404120 1.0212337 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.4039357
1897 0.4783966 -0.0817113 -1.1926732 -0.7589986 -0.6741149 1.0212337 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.3931991
1901 -0.6109590 -1.2710382 -0.9472574 -0.7589986 2.6404120 1.0212337 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.5007895
1910 -0.0740747 0.6794579 -0.6356183 -0.0501185 0.4307274 1.0212337 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.2503772
1919 -0.6386691 -1.2710382 -0.9472574 -0.0501185 -0.6741149 -1.7213059 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4423148
1922 -0.7321909 -0.8904536 -0.9472574 -0.7589986 0.4307274 -1.7213059 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.4633848
1929 -0.7408504 -0.8904536 -1.1926732 -1.1134387 0.4307274 1.0212337 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.3560106
1930 -0.9053794 -0.8904536 -0.6356183 -0.7589986 -0.6741149 1.0212337 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.1838114
1937 1.0862882 0.6794579 -0.2850242 -0.0501185 2.6404120 1.0212337 C96 2.0782294 0.7478725 0.1122044 0.7127714 2.6404120 0.5611990 0.4467515
1940 1.3529985 0.6794579 0.4940736 -0.0501185 0.4307274 1.0212337 C219 1.9755430 0.5864499 0.2031375 -0.2014525 0.1203785 0.5809108 0.3180824
1941 2.2120134 -0.0817113 0.1512706 -0.0501185 -0.6741149 1.0212337 C219 1.9755430 0.5864499 0.2031375 -0.2014525 0.1203785 0.5809108 0.3904415
1944 0.6446576 -1.2710382 0.4940736 1.3676416 -0.6741149 1.0212337 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.2400317
1965 0.1337515 0.6794579 -0.6356183 -0.0501185 0.4307274 1.0212337 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.2157395
1971 -1.2604158 -1.2710382 -1.1926732 -1.1134387 0.4307274 1.0212337 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.3324035
1975 1.3443391 -0.8904536 0.4940736 1.3676416 -0.6741149 1.0212337 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.2232783
1979 -2.0241771 -0.8904536 -1.2978514 -1.1134387 -0.6741149 1.0212337 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.3023442
1986 -0.9053794 -0.8904536 -0.9472574 -0.7589986 -0.6741149 -1.7213059 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3005997
1998 0.6533170 -0.8904536 0.4940736 1.3676416 -0.6741149 1.0212337 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.1751576
2007 -0.5607343 -0.8904536 -1.1926732 -1.1134387 -0.6741149 1.0212337 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.2117200
2010 0.1406791 0.6794579 -1.1420319 -0.7589986 -0.6741149 1.0212337 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.2961620
2014 -1.2517564 -0.8904536 -1.1926732 -1.1134387 -0.6741149 1.0212337 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.1624117
2015 0.8247736 -1.2710382 0.4940736 1.3676416 -0.6741149 1.0212337 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.2100123
2019 -1.2448289 -0.8904536 -1.1030770 -0.7589986 -0.6741149 1.0212337 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.1137335
2020 -0.3858139 -1.2710382 0.0499878 -0.0501185 -0.6741149 1.0212337 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.2066937
2023 3.0848834 0.6794579 0.1434796 -0.0501185 2.6404120 1.0212337 C96 2.0782294 0.7478725 0.1122044 0.7127714 2.6404120 0.5611990 0.3882114
2025 3.2511444 0.6794579 0.3382540 1.3676416 2.6404120 1.0212337 C96 2.0782294 0.7478725 0.1122044 0.7127714 2.6404120 0.5611990 0.4303807
2040 -1.0716404 0.6794579 -1.1030770 -0.7589986 -0.6741149 1.0212337 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.2782722
2048 1.4309333 -0.0817113 -0.6550957 -0.0501185 -0.6741149 -1.7213059 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.3679198
2054 1.3374115 -0.0817113 0.4940736 1.3676416 -0.6741149 1.0212337 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.3344434
2056 -0.3511762 0.6794579 -0.7914378 -0.7589986 -0.6741149 1.0212337 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.1729719
2061 3.7707099 -0.0817113 2.6755476 1.3676416 -0.6741149 1.0212337 C55 1.4810668 0.0181922 2.3175726 2.6287019 0.1225346 -0.2170130 1.0072463
2066 0.8334330 -0.8904536 -0.2655468 1.3676416 0.4307274 1.0212337 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.3620957
2071 -0.5590024 -0.8904536 -0.7836469 -0.0501185 0.4307274 1.0212337 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.2243028
2090 1.4309333 -0.8904536 0.4940736 1.3676416 -0.6741149 1.0212337 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.2377106
2093 0.6602445 -0.0817113 -0.7135281 -0.7589986 -0.6741149 1.0212337 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.3868376
2094 -0.8118576 -0.8904536 -0.7135281 -0.7589986 -0.6741149 1.0212337 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1656158
2097 -0.5607343 -1.2710382 -0.7758559 -0.0501185 -0.6741149 1.0212337 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.2660686
2102 1.0066215 0.6794579 0.1434796 -0.0501185 0.4307274 1.0212337 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.2771260
2103 -1.0196838 -1.2710382 -1.2978514 -0.0501185 -0.6741149 -1.7213059 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.5204847
2106 -0.3511762 -0.8904536 -0.6356183 -0.0501185 0.4307274 1.0212337 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.2055441
2110 1.1728825 -0.0817113 0.4940736 1.3676416 -0.6741149 1.0212337 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.3070218
2113 0.1250921 0.6794579 -0.7758559 -0.7589986 0.4307274 1.0212337 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.3213397
2114 -0.3858139 -0.8904536 -0.7758559 -0.7589986 -0.6741149 1.0212337 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1265840
2116 2.3852019 0.6794579 0.1512706 -0.0501185 -0.6741149 1.0212337 C219 1.9755430 0.5864499 0.2031375 -0.2014525 0.1203785 0.5809108 0.3234473
2118 -0.4741401 -0.0817113 -1.1926732 -1.1134387 -0.6741149 1.0212337 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.2650090
2123 0.2445922 -0.8904536 -0.2850242 -0.0501185 2.6404120 1.0212337 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.2192632
2124 0.9113679 0.6794579 0.0499878 -0.0501185 -0.6741149 1.0212337 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.2297720
2126 0.1337515 -0.0817113 -0.2850242 -0.0501185 0.4307274 0.3221550 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.2205069
2129 -0.2056979 -0.8904536 -1.1420319 -0.7589986 -0.6741149 0.3221550 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1522240
2133 0.6533170 -1.2710382 -0.6550957 -0.0501185 -0.6741149 -1.7213059 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4168555
2134 0.1337515 -0.8904536 0.0499878 -0.0501185 -0.6741149 0.3221550 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.0905453
2142 -1.2604158 -1.2710382 -1.1926732 -1.1134387 -0.6741149 0.3221550 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.2370416
2145 0.4870561 -0.0817113 -0.2850242 -0.0501185 0.4307274 0.3221550 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.2420173
2146 -0.3788864 0.6794579 -0.9472574 -0.7589986 -0.6741149 0.3221550 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.1739817
2149 0.6515851 -0.0817113 0.0499878 -0.0501185 -0.6741149 0.3221550 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.2315532
2152 -0.8118576 -0.0817113 -0.7836469 -0.7589986 -0.6741149 0.3221550 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.2307387
2156 -0.5001183 0.6794579 -0.7914378 -0.7589986 0.4307274 0.3221550 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.2927816
2157 -0.4221835 -1.2710382 -1.1420319 -0.7589986 -0.6741149 0.3221550 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1520934
2158 -0.7079445 -0.0817113 -0.7914378 -0.7589986 0.4307274 0.3221550 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.3177012
2161 0.0454254 -0.8904536 0.0499878 -0.0501185 -0.6741149 0.3221550 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.0758243
2164 -0.3788864 -0.8904536 -1.1420319 -0.0501185 -0.6741149 -1.7213059 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3680494
2179 -1.4249449 -1.2710382 -0.9472574 -0.7589986 0.4307274 0.3221550 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.3429802
2182 -0.0325094 -0.0817113 -0.7135281 -0.0501185 0.4307274 0.3221550 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.3196344
2183 -0.9053794 -0.8904536 -1.1926732 -1.1134387 -0.6741149 0.3221550 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.2339490
2188 -0.6473286 -1.2710382 -0.7758559 -0.7589986 -0.6741149 0.3221550 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1285448
2190 3.2511444 0.6794579 0.3382540 1.3676416 2.6404120 0.3221550 C96 2.0782294 0.7478725 0.1122044 0.7127714 2.6404120 0.5611990 0.3935489
2193 -0.9053794 -1.2710382 -0.7758559 -0.7589986 -0.6741149 0.3221550 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1715532
2205 0.4350995 -0.8904536 -0.6550957 -0.0501185 -0.6741149 0.3221550 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.2582839
2207 -0.3009516 -0.8904536 -0.7758559 -0.7589986 -0.6741149 0.3221550 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1055158
2209 0.0454254 -0.0817113 -0.7758559 -0.7589986 -0.6741149 0.3221550 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.2708561
2215 0.6256068 -0.8904536 -0.2850242 -0.7589986 -0.6741149 0.3221550 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.3224593
2218 0.3502371 -0.8904536 -0.9472574 -0.0501185 0.4307274 -1.7213059 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4077083
2221 0.2186139 -0.8904536 -1.1420319 -0.7589986 -0.6741149 0.3221550 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.2229426
2228 1.6907161 0.6794579 0.0421969 1.3676416 0.4307274 0.3221550 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.3025392
2230 0.1926356 -0.8904536 -0.2850242 -0.0501185 2.6404120 0.3221550 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.1993018
2235 -1.0872273 -0.8904536 -1.1926732 -1.1134387 0.4307274 0.3221550 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.3904060
2248 0.4853242 -0.0817113 -0.7135281 -0.7589986 -0.6741149 0.3221550 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.3545605
2254 -1.0716404 -1.2710382 -1.1420319 -0.7589986 -0.6741149 0.3221550 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.2251638
2259 -0.8984519 0.6794579 -0.9472574 -0.7589986 -0.6741149 0.3221550 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.2203138
2261 -2.1107713 -0.8904536 -1.2978514 -1.1134387 -0.6741149 0.3221550 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.3791820
2263 -0.6473286 -0.8904536 -1.1926732 -1.1134387 -0.6741149 0.3221550 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1909405
2264 0.4714691 -0.0817113 0.0499878 -0.0501185 -0.6741149 0.3221550 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.2015339
2276 0.6533170 -1.2710382 -0.2850242 1.3676416 -0.6741149 -1.7078621 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.6314472
2283 4.8167683 0.6794579 0.4356413 -0.0501185 2.6404120 0.3221550 C96 2.0782294 0.7478725 0.1122044 0.7127714 2.6404120 0.5611990 0.6887207
2289 1.3374115 -0.0817113 0.4473277 1.3676416 0.4307274 0.3221550 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.3305615
2295 1.2577448 -0.8904536 0.4473277 1.3676416 -0.6741149 0.3221550 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.1505595
2296 0.6446576 -1.2710382 0.4473277 1.3676416 -0.6741149 0.3221550 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.1817453
2300 -0.7772199 -0.8904536 -0.9472574 -0.7589986 -0.6741149 0.3221550 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1126131
2303 1.3460710 0.6794579 0.0421969 1.3676416 0.4307274 0.3221550 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.2450983
2307 0.8178461 -0.0817113 -0.2850242 -0.0501185 0.4307274 0.3221550 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.2971490
2308 1.1798100 -0.0817113 -0.3629340 -0.0501185 0.4307274 0.3221550 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.3704613
2316 1.6041218 -0.0817113 0.4473277 1.3676416 -0.6741149 0.3221550 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.3206087
2321 1.3460710 0.6794579 0.4473277 1.3676416 -0.6741149 0.3221550 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.2430285
2323 -0.9053794 -0.8904536 -0.9472574 -0.7589986 -0.6741149 -1.7078621 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.2983591
2326 -0.2143573 0.6794579 -1.1926732 -1.1134387 -0.6741149 0.3221550 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.2728719
2331 1.5106000 0.6794579 0.4473277 1.3676416 0.4307274 0.3221550 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.2325647
2332 0.8178461 -0.8904536 0.4473277 1.3676416 0.4307274 0.3221550 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.2046355
2333 1.1728825 -0.0817113 0.4473277 1.3676416 -0.6741149 0.3221550 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.2487355
2334 -0.5676619 -0.8904536 -0.7758559 -0.7589986 -0.6741149 0.3221550 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.0793166
2335 0.6533170 -0.8904536 0.4473277 1.3676416 -0.6741149 0.3221550 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.1168713
2336 -0.2126255 -0.8904536 -0.2850242 -0.0501185 0.4307274 0.3221550 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.1953703
2340 5.5095223 0.6794579 3.0650965 4.2031619 2.6404120 0.8195764 B1 1.0281464 1.1551887 3.0650965 2.7854017 2.6404120 -2.3416423 1.5893476
2351 -0.2212849 -0.8904536 -0.7758559 -0.7589986 0.4307274 0.8195764 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.1911115
2362 -0.0394370 -0.8904536 -0.7836469 -0.7589986 2.6404120 0.8195764 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.2812271
2364 1.3529985 0.6794579 -0.3239791 -0.7589986 0.4307274 -1.7078621 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.4510968
2380 0.6446576 0.6794579 0.0499878 -0.0501185 0.4307274 0.8195764 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.1676072
2396 -0.8811330 -1.2710382 -1.1420319 -0.7589986 -0.6741149 0.8195764 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.1740115
2401 0.2186139 -0.0817113 -0.2850242 -0.0501185 0.4307274 0.8195764 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.1781350
2402 -0.0480964 -0.8904536 0.0499878 -0.0501185 -0.6741149 0.8195764 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.0756235
2404 2.0215061 0.6794579 3.7273297 -0.0501185 -0.6741149 0.8195764 C219 1.9755430 0.5864499 0.2031375 -0.2014525 0.1203785 0.5809108 0.8079427
2409 -1.4249449 -0.8904536 -0.7836469 -0.7589986 -0.6741149 0.8195764 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.1034432
2410 0.9113679 -1.2710382 0.4473277 1.3676416 -0.6741149 0.8195764 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.1731466
2419 0.1337515 -0.8904536 0.0499878 -0.0501185 -0.6741149 0.8195764 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.1018743
2420 -0.8984519 0.6794579 -0.9472574 -0.7589986 -0.6741149 0.8195764 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.1898280
2429 -1.4249449 -1.2710382 -1.1926732 -1.1134387 -0.6741149 0.8195764 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.1781120
2436 -0.8620823 -1.2710382 -0.9472574 -0.7589986 -0.6741149 -1.7213059 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3712467
2440 1.3460710 0.6794579 -0.6356183 -0.0501185 2.6404120 -1.7078621 C142 0.1328959 0.6627507 0.4814828 -0.0248014 2.6404120 -0.9712527 0.5181517
2442 0.2532516 -0.8904536 -0.7836469 -0.7589986 -0.6741149 0.8195764 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1981869
2443 0.6446576 -1.2710382 0.4473277 1.3676416 -0.6741149 0.8195764 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.1986311
2452 0.2705704 -0.8904536 -1.1420319 -0.7589986 -0.6741149 0.8195764 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.2332044
2454 -0.0480964 -0.0817113 -0.7758559 -0.7589986 -0.6741149 0.8195764 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.2247832
2456 -0.9053794 -1.2710382 -0.7758559 -0.7589986 -0.6741149 0.8195764 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.1536080
2458 -0.2126255 0.6794579 -0.2850242 -0.7589986 0.4307274 0.8195764 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.2622111
2462 -0.3944734 0.6794579 -1.1926732 -1.1134387 -0.6741149 0.8195764 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.2123667
2472 0.1320196 -0.8904536 0.0499878 -0.0501185 -0.6741149 0.8195764 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.1015856
2479 1.6041218 0.6794579 0.4473277 1.3676416 -0.6741149 0.8195764 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.2438723
2491 0.3069400 -0.0817113 -0.2850242 -0.7589986 2.6404120 0.8195764 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.1853942
2493 -0.1260312 0.6794579 -0.9472574 -0.7589986 -0.6741149 -1.7078621 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3903145
2494 -0.7252634 -0.8904536 -1.1420319 -0.7589986 -0.6741149 0.8195764 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1380184
2498 -0.9937055 -1.2710382 -1.1926732 -1.1134387 -0.6741149 0.8195764 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.1985897
2510 0.3918024 -0.8904536 0.0499878 -0.0501185 -0.6741149 0.8195764 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.1448827
2532 0.8438243 0.6794579 0.5135511 -0.0501185 0.4307274 0.8195764 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.2780622
2540 -1.5981334 -1.2710382 -0.9472574 -0.7589986 -0.6741149 0.8195764 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.1311740
2549 -0.2126255 -0.0817113 -0.7758559 -0.7589986 -0.6741149 0.8195764 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.1973617
2551 0.1320196 -0.0817113 -0.7758559 -0.7589986 -0.6741149 0.8195764 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.2548026
2555 0.9633244 -0.8904536 -0.2850242 -0.7589986 -0.6741149 0.8195764 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.3900745
2565 -0.7079445 -0.0817113 -0.7914378 -0.7589986 0.4307274 0.8195764 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.3074258
2566 -0.7495098 -0.8904536 -0.2850242 -0.7589986 0.4307274 0.8195764 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.1619970
2568 1.3460710 0.6794579 -0.2850242 -0.0501185 0.4307274 -1.7078621 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.3276122
2569 1.6907161 0.6794579 0.4473277 1.3676416 -0.6741149 0.8195764 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.2583046
2573 1.5106000 0.6794579 0.4473277 1.3676416 -0.6741149 0.8195764 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.2282853
2575 0.9113679 -0.8904536 0.4473277 1.3676416 -0.6741149 0.8195764 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.1097159
2582 0.8853896 0.6794579 -0.6550957 -0.7589986 0.4307274 0.8195764 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.3540595
2591 0.3069400 0.6794579 -0.2850242 -0.0501185 0.4307274 0.8195764 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.1151887
2595 -1.2604158 -1.2710382 -1.1926732 -1.1134387 -0.6741149 0.8195764 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.1541380
2601 -0.5520749 -0.0817113 -0.2850242 -0.7589986 -0.6741149 0.8195764 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.2400595
2603 0.0454254 -0.8904536 0.0499878 -0.0501185 -0.6741149 0.8195764 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.0871532
2604 -0.5590024 0.6794579 -1.1926732 -1.1134387 -0.6741149 0.8195764 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.2047218
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2609 -1.2448289 -1.2710382 -0.9472574 -0.7589986 -0.6741149 -1.7078621 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3258541
2611 2.4787237 0.6794579 0.1434796 -0.0501185 0.4307274 0.8195764 C219 1.9755430 0.5864499 0.2031375 -0.2014525 0.1203785 0.5809108 0.2260325
2612 -1.4180174 -0.8904536 -0.7836469 -0.7589986 -0.6741149 0.8195764 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.1022886
2613 -0.3858139 0.6794579 -0.2850242 -0.7589986 -0.6741149 0.8195764 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.2179917
2614 0.7901359 -0.8904536 -0.6550957 -0.0501185 -0.6741149 0.8195764 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.3287856
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4728 -0.8205171 0.6794579 -1.1926732 -1.1134387 -0.6741149 1.5724304 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.3737832
4732 0.7087373 2.2969425 0.4940736 -0.0501185 0.4307274 1.5724304 C301 0.3539798 2.2969425 -0.1275540 -0.2934206 -0.2808659 0.8171839 0.4477545
4734 0.1250921 0.6794579 0.4473277 -0.0501185 -0.6741149 1.5724304 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.3510786
4736 -1.0785679 0.6794579 -0.7758559 -0.7589986 -0.6741149 1.5724304 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.3391682
4748 1.7513320 0.6794579 0.4434322 -0.0501185 -0.6741149 1.5724304 C219 1.9755430 0.5864499 0.2031375 -0.2014525 0.1203785 0.5809108 0.4158101
4749 -0.9140388 -0.8904536 -1.2783740 -1.1134387 -0.6741149 -1.5330924 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3820470
4752 0.9996940 1.1076156 0.4473277 1.3676416 -0.6741149 1.5724304 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.3440322
4753 -0.3858139 -0.8904536 0.0460923 -0.0501185 0.4307274 1.5724304 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.3368919
4757 -1.4249449 -0.8904536 -0.7758559 -0.7589986 -0.6741149 1.5724304 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.2171059
4758 0.1753168 -0.8904536 -0.2850242 -0.7589986 -0.6741149 1.5724304 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.3842156
4769 1.2317666 0.6794579 0.4434322 1.3676416 0.4307274 1.5724304 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.2988757
4787 -0.9140388 0.6794579 -1.1926732 -1.1134387 -0.6741149 1.5724304 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.3893702
4788 1.1278535 0.6794579 -0.2850242 -0.0501185 -0.6741149 1.5724304 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.3716236
4790 -0.5849807 2.2969425 -0.7914378 -0.7589986 0.4307274 1.5724304 C301 0.3539798 2.2969425 -0.1275540 -0.2934206 -0.2808659 0.8171839 0.5892104
4793 -0.7408504 -0.0817113 -0.7758559 -0.7589986 0.4307274 1.5724304 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.4357887
4796 -0.2212849 -0.8904536 0.0499878 -0.0501185 0.4307274 1.5724304 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.3649627
4800 1.5106000 0.6794579 -0.2850242 1.3676416 -0.6741149 -1.5330924 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.3390723
4801 -0.5607343 -0.0817113 0.0266149 -0.7589986 0.4307274 1.5724304 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.4244049
4805 -0.1294950 -0.8904536 0.0499878 -0.0501185 -0.6741149 1.5724304 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.2146656
4810 -0.0480964 2.2969425 -0.7758559 -0.7589986 -0.6741149 0.6851382 C301 0.3539798 2.2969425 -0.1275540 -0.2934206 -0.2808659 0.8171839 0.3402084
4813 -0.3788864 -0.8904536 0.4434322 -0.7589986 -0.6741149 0.6851382 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.2678298
4817 -1.4526551 -0.8904536 -0.2850242 -0.7589986 -0.6741149 0.6851382 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.2135717
4825 0.0436935 0.6794579 0.0499878 -0.0501185 -0.6741149 0.6851382 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.1505396
4832 -0.2160892 -0.8904536 0.0499878 -0.0501185 -0.6741149 0.6851382 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.0812160
4841 -1.0266114 -0.8904536 -0.6745732 -0.7589986 -0.6741149 0.6851382 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1518846
4842 -0.9071113 -0.8904536 -0.7758559 -0.7589986 -0.6741149 0.6851382 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1150875
4843 0.2203458 -0.8904536 0.4473277 1.3676416 -0.6741149 0.6851382 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.1835126
4845 0.6602445 -0.8904536 -0.6745732 -0.7589986 0.4307274 -1.5330924 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4203032
4846 0.8265055 0.6794579 0.4940736 1.3676416 0.4307274 0.6851382 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.2310513
4848 0.3069400 2.2969425 0.0499878 -0.0501185 -0.6741149 0.6851382 C301 0.3539798 2.2969425 -0.1275540 -0.2934206 -0.2808659 0.8171839 0.1655297
4849 -0.2645820 -0.8904536 0.4940736 -0.0501185 0.4307274 0.6851382 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.2838787
4853 -0.2160892 -0.8904536 0.0499878 -0.0501185 -0.6741149 0.6851382 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.0812160
4858 1.0066215 0.6794579 0.4434322 1.3676416 0.4307274 0.6851382 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.1925917
4859 1.5192595 0.6794579 2.2859986 2.7854017 -0.6741149 0.6851382 C55 1.4810668 0.0181922 2.3175726 2.6287019 0.1225346 -0.2170130 0.4310888
4862 0.4870561 0.6794579 -0.2850242 -0.7589986 -0.6741149 0.6851382 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.1836006
4864 -0.3078791 -0.8904536 0.0499878 -0.0501185 -0.6741149 0.6851382 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.0965143
4865 1.3529985 2.2969425 0.4473277 1.3676416 2.6404120 0.6851382 C96 2.0782294 0.7478725 0.1122044 0.7127714 2.6404120 0.5611990 0.5647056
4878 -0.8984519 -0.0817113 -0.2850242 -0.0501185 0.4307274 0.6851382 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.3067138
4886 -0.0394370 -0.0817113 -1.1030770 -0.7589986 -0.6741149 -1.5330924 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2458622
4887 -1.0716404 -0.8904536 -0.7914378 -0.7589986 -0.6741149 0.6851382 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.1320396
4897 -1.5912058 -0.8904536 -0.7914378 -0.7589986 -0.6741149 0.6851382 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.1522612
4907 -0.1260312 0.6794579 0.4940736 -0.0501185 0.4307274 0.6851382 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.2147349
4916 1.2594767 -0.8904536 2.2859986 2.7854017 -0.6741149 0.6851382 C55 1.4810668 0.0181922 2.3175726 2.6287019 0.1225346 -0.2170130 0.5028851
4919 0.5840416 0.6794579 0.4940736 -0.0501185 0.4307274 0.6851382 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.2091124
4928 0.3623603 0.6794579 0.4940736 -0.0501185 0.4307274 0.6851382 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.1721656
4938 0.9113679 0.6794579 0.4473277 1.3676416 -0.6741149 0.6851382 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.2470019
4947 -0.1537414 0.6794579 0.0266149 -0.0501185 -0.6741149 0.6851382 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.1795500
4958 0.4870561 2.2969425 0.0577788 1.3676416 0.4307274 0.6851382 C301 0.3539798 2.2969425 -0.1275540 -0.2934206 -0.2808659 0.8171839 0.4705184
4959 0.9996940 2.2969425 0.4473277 1.3676416 -0.6741149 0.6851382 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.3984452
4964 0.6533170 -0.0817113 -0.6356183 -0.0501185 0.4307274 -1.5330924 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3937154
4974 -0.0325094 0.6794579 0.0266149 -0.0501185 -0.6741149 0.6851382 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.1593446
4976 -0.9919737 -0.8904536 -0.7758559 -0.7589986 -0.6741149 0.6851382 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1292312
4989 -0.8187852 0.6794579 0.0460923 -0.0501185 0.4307274 0.6851382 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.2555303
4999 0.8143823 -0.8904536 -0.7836469 -0.7589986 2.6404120 0.6851382 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.3094387
5005 0.3848748 -0.8904536 0.4473277 1.3676416 -0.6741149 0.6851382 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.1560911
5007 1.2560130 -0.8904536 2.2859986 2.7854017 -0.6741149 0.6851382 C55 1.4810668 0.0181922 2.3175726 2.6287019 0.1225346 -0.2170130 0.5034623
5017 0.1406791 0.6794579 0.0577788 1.3676416 0.4307274 0.6851382 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.3338607
5021 0.7225924 0.6794579 2.5976378 -0.0501185 0.4307274 0.6851382 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.5827983
5022 -0.0480964 0.6794579 0.0499878 -0.0501185 0.4307274 0.6851382 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.1277315
5029 -1.0785679 0.6794579 -0.7758559 -0.7589986 -0.6741149 0.6851382 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.1912861
5033 0.5147662 -0.8904536 -0.6356183 -0.0501185 0.4307274 -1.5330924 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4004619
5044 1.4309333 -0.8904536 2.2859986 2.7854017 -0.6741149 0.6851382 C55 1.4810668 0.0181922 2.3175726 2.6287019 0.1225346 -0.2170130 0.4743090
5048 0.4714691 2.2969425 0.4473277 -0.0501185 -0.6741149 0.6851382 C301 0.3539798 2.2969425 -0.1275540 -0.2934206 -0.2808659 0.8171839 0.2434946
5057 -0.7339228 -0.8904536 -0.2850242 -0.7589986 -0.6741149 0.6851382 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1680280
5060 -1.2448289 -0.8904536 -0.7914378 -0.7589986 0.4307274 0.6851382 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.2580452
5064 0.6931504 0.6794579 0.0577788 1.3676416 2.6404120 0.6851382 C96 2.0782294 0.7478725 0.1122044 0.7127714 2.6404120 0.5611990 0.3811214
5084 0.5494039 -0.8904536 2.5976378 -0.0501185 0.4307274 0.6851382 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.7508350
5096 0.0869906 -0.8904536 -0.7836469 -0.7589986 0.4307274 0.6851382 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.2213829
5106 -1.4180174 -0.0817113 -0.2850242 -0.7589986 -0.6741149 0.6851382 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.3425891
5110 -0.4758720 -0.8904536 0.0499878 -0.0501185 -0.6741149 0.6851382 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.1245131
5113 -0.1727921 -0.0817113 -0.6356183 -0.7589986 2.6404120 0.6851382 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.2477972
5116 -0.2472631 0.6794579 -0.9472574 -0.7589986 -0.6741149 -1.5330924 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3813915
5129 -1.4249449 -0.8904536 -0.7758559 -0.7589986 -0.6741149 0.6851382 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.1271480
5130 0.4524184 2.2969425 0.0266149 -0.0501185 0.4307274 0.6851382 C301 0.3539798 2.2969425 -0.1275540 -0.2934206 -0.2808659 0.8171839 0.2232581
hist(sorted_df$diff, breaks = 20, col = "blue", main = "Mean Absolute Difference", xlab = "Difference")
Figure 22: Mean Absolute Difference

Figure 22: Mean Absolute Difference

7 Executive Summary

8 References

  1. Topology Preserving Maps : https://users.ics.aalto.fi/jhollmen/dippa/node9.html#:~:text=The%20property%20of%20topology%20preserving,tool%20of%20high-dimensional%20data

  2. Vector Quantization : https://en.wikipedia.org/wiki/Vector_quantization

  3. K-means : https://en.wikipedia.org/wiki/K-means_clustering

  4. Sammon’s Projection : http://en.wikipedia.org/wiki/Sammon_mapping

  5. Voronoi Tessellations : http://en.wikipedia.org/wiki/Centroidal_Voronoi_tessellation